(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 171467, 3994]*) (*NotebookOutlinePosition[ 172112, 4016]*) (* CellTagsIndexPosition[ 172068, 4012]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[BoxData[{ \(\(SetDirectory["\"];\)\), "\[IndentingNewLine]", \(<< RecursionOperator.m\)}], "Input"], Cell[BoxData[ \("Package InvariantsSymmetries has been successfully loaded"\)], "Print"], Cell[BoxData[ \("The Recursion Operator package loaded successfully."\)], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Burgers'\ Equation\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], \ t]\ == u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 2}], \[IndentingNewLine]u[x, t], {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[3]D_x^{1} + C[3]uI + C[3]u_{x}D_x^{-1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\((2\ C[3])\)*operatorD[1]\), "+", \(C[3]*u[x, t]*operatorD[0]\), "+", RowBox[{\(C[3]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Burgers'\ Equation\ with\ explicit\ dependence\ on\ x\ and\ \ \(\(t\)\(.\)\)\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], \ t]\ == u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 2}], \[IndentingNewLine]u[x, t], {x, t}, MaxExplicitDependency \[Rule] 1]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[5]tD_x^{1} + C[5]D_x^{-1} + C[5]tuI + C[5]tu_{x}D_x^{-1} + \ C[5]xI}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(C[5]*operatorD[\(-1\)]\), "+", \(C[5]*x*operatorD[0]\), "+", \(\((2\ C[5])\)*t*operatorD[1]\), "+", \(C[5]*t*u[x, t]*operatorD[0]\), "+", RowBox[{\(C[5]\), "*", "t", "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Burger' s\ Equation\ with\ parameter\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\[IndentingNewLine]D[u[x, t], t] \[Equal] \(-\ u[x, t]\)*D[u[x, t], x]\ + \ \[Nu]* D[u[x, t], \ {x, 2}], \[IndentingNewLine]u[x, t], {x, t}, {\[Nu]}]\)\)\)], "Input"], Cell[BoxData[ \("{{{D_x^{1} + \\frac{-1}{2} \\frac{1}{\[Nu]^{1}}uI + \\frac{-1}{2} \ \\frac{1}{\[Nu]^{1}}u_{x}D_x^{-1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(DefiningEquation`Operator`Times[operatorD[1]]\), "+", \(\(-\(1\/\(2\ \[Nu]\)\)\)*u[x, t]*operatorD[0]\), "+", RowBox[{\(-\(1\/\(2\ \[Nu]\)\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Potential\ Burgers'\ \(Equation\ --\)\ R\ is\ only\ \(\(local\)\(.\)\)\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ D[u[x, t], x]^2 + D[u[x, t], {x, 2}], u[x, t], {x, t}, \ WeightRules\ \[Rule] \ {weight[u] \[Rule] 1}, Seed \[Rule] 2]\)\)\)], "Input"], Cell[BoxData[ \(weight::"nonuniform1" \(\(:\)\(\ \)\) "Given system has at least one equation with terms of unequal rank, so \ that scaling properties can not be determined."\)], "Message"], Cell[BoxData[ \(weight::"nonuniform2" \(\(:\)\(\ \)\) "Incompatibility has been cured by assuming auxiliary parameters with \ weight."\)], "Message"], Cell[BoxData[ \("{{{C[1]I + C[7]D_x^{1} + C[7]u_{x}I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(C[1]*operatorD[0]\), "+", \(C[7]*operatorD[1]\), "+", RowBox[{\(C[7]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Potential\ Burgers'\ Equation\ with\ explicit\ dependence\ on\ x\ and\ \ \(t\ --\)\ R\ is\ only\ \(\(local\)\(.\)\)\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ D[u[x, t], x]^2 + D[u[x, t], {x, 2}], u[x, t], {x, t}, WeightRules \[Rule] {weight[u] \[Rule] \(-weight[x]\)}, Seed \[Rule] 2, MaxExplicitDependency\ \[Rule] 1]\)\)\)], "Input"], Cell[BoxData[ \(weight::"nonuniform1" \(\(:\)\(\ \)\) "Given system has at least one equation with terms of unequal rank, so \ that scaling properties can not be determined."\)], "Message"], Cell[BoxData[ \(weight::"nonuniform2" \(\(:\)\(\ \)\) "Incompatibility has been cured by assuming auxiliary parameters with \ weight."\)], "Message"], Cell[BoxData[ \("{{{C[8]tD_x^{1} + C[8]tu_{x}I + \\frac{1}{2} C[8]xI}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(C[8]\/2*x*operatorD[0]\), "+", \(C[8]*t*operatorD[1]\), "+", RowBox[{\(C[8]\), "*", "t", "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[BoxData[ \(\(\( (*\ Diffusion\ Equation\ *) \)\(\[IndentingNewLine]\)\( (*\ R1\ is\ composed\ of\ the\ nonpolynomial\ \[Rho]\ = \ 1/u\ \((or\ cosymmetry\ 1/u^2)\)\ *) \)\(\[IndentingNewLine]\)\( (*\ RecursionOperator[\[IndentingNewLine]D[u[x, t], t]\ \[Equal] \ u[x, t]^2*D[u[x, t], {x, 2}], \[IndentingNewLine]u[x, t], {x, t}, \ WeightRules \[Rule] {weight[ u] \[Rule] \(-weight[ x]\)}]\[IndentingNewLine]*) \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ The\ Korteweg - de\ Vries\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], t]\ \[Equal] \ 6*u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 3}], \[IndentingNewLine]u[x, t], {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[3]uI + C[3]u_{x}D_x^{-1} + \\frac{1}{2} C[3]D_x^{2}}}}"\)], \ "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(C[3]\/2*operatorD[2]\), "+", RowBox[{\(C[3]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[3])\)*u[x, t]*operatorD[0]\)}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Potential\ Korteweg - de\ Vries\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], t]\ \[Equal] \ 3*D[u[x, t], x]^2 + D[u[x, t], {x, 3}], u[x, t], {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{-2C[5]D_x^{-1}u_{2 x}I + 4 C[5]u_{x}I + C[5]D_x^{2}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(C[5]*operatorD[2]\), "+", RowBox[{\((4\ C[5])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \(C[5]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Modified\ Korteweg - de\ Vries\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], t]\ \[Equal] \ u[x, t]^2*D[u[x, t], x] + D[u[x, t], {x, 3}], \[IndentingNewLine]u[x, t], {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[5]u^{2}I + 2C[5]u_{x}D_x^{-1}uI + 3 C[5]D_x^{2}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\((3\ C[5])\)*operatorD[2]\), "+", \(\((2\ C[5])\)*u[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(C[5]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Potential\ Modified\ Korteweg - de\ Vries\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], \ t]\ \[Equal] \ 1/3*D[u[x, t], x]^3 + D[u[x, t], {x, 3}], \[IndentingNewLine]u[x, t], {x, t}, \ \[IndentingNewLine]WeightRules\ \[Rule] \ {weight[ u] \[Rule] 1}, \[IndentingNewLine]Seed\ \[Rule] \ 2\[IndentingNewLine]]\)\)\)], "Input"], Cell[BoxData[ \(weight::"nonuniform1" \(\(:\)\(\ \)\) "Given system has at least one equation with terms of unequal rank, so \ that scaling properties can not be determined."\)], "Message"], Cell[BoxData[ \(weight::"nonuniform2" \(\(:\)\(\ \)\) "Incompatibility has been cured by assuming auxiliary parameters with \ weight."\)], "Message"], Cell[BoxData[ \("{{{2 C[19]u_{x}^{2}I + -2C[19]u_{x}D_x^{-1}u_{2 x}I + 3 C[19]D_x^{2} + \ C[1]I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(C[1]*operatorD[0]\), "+", \(\((3\ C[19])\)*operatorD[2]\), "+", RowBox[{\((2\ C[19])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \(C[19]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[BoxData[ \(\(\( (*\ The\ Cylindrical\ Korteweg - de\ Vries\ *) \)\(\[IndentingNewLine]\)\( (*\ R1\ is\ not\ the\ composition\ of\ symmetries\ and\ \ \(\(cosymmetries\)\(.\)\)\ *) \)\(\[IndentingNewLine]\)\( (*\ RecursionOperator[\[IndentingNewLine]D[u[x, t], t]\ \[Equal] \ u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 3}] - u[x, t]/\((2*t)\), \[IndentingNewLine]u[x, t], {x, t}, MaxExplicitDependency\ \[Rule] \ 1]\[IndentingNewLine]*) \)\)\)], "Input"], Cell[BoxData[ \(\(\( (*\ Harry\ Dym\ Equation\ *) \)\(\[IndentingNewLine]\)\( (*\ R1\ composed\ of\ nonpolynomial\ density\ \[Rho]\ = \ 1/\(\(u\)\(.\)\)\ *) \)\(\[IndentingNewLine]\)\( (*RecursionOperator[\ \[IndentingNewLine]D[u[x, t], t]\ \[Equal] \ u[x, t]^3*D[u[x, t], {x, 3}], \[IndentingNewLine]u[x, t], {x, t}, \ WeightRules \[Rule] {weight[u] \[Rule] 1}]\[IndentingNewLine]*) \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Kupershmidt\ Equation\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], t] \[Equal] D[u[x, t], {x, 5}] + 5*D[u[x, t], x]*D[u[x, t], {x, 3}] + 5*D[u[x, t], {x, 2}]^2 - 5*u[x, t]^2*D[u[x, t], {x, 3}] - 20*u[x, t]*D[u[x, t], x]*D[u[x, t], {x, 2}] - 5*D[u[x, t], x]^3 + 5*u[x, t]^4*D[u[x, t], x], \[IndentingNewLine]u[x, t], {x, t}, Seed \[Rule] 2]\)\)\)], "Input"], Cell[BoxData[ \("{{{-10C[42]u^{2}u_{3 x}D_x^{-1}uI + 10C[42]u_{2 x}^{2}D_x^{-1}uI + \ 10C[42]u^{4}u_{x}D_x^{-1}uI + -10C[42]u_{x}^{3}D_x^{-1}uI + \ -10C[42]u_{x}D_x^{-1}u^{2}u_{2 x}I + -10C[42]u_{x}D_x^{-1}uu_{x}^{2}I + \ 10C[42]u_{x}D_x^{-1}u_{x}u_{2 x}I + 10C[42]u_{x}u_{3 x}D_x^{-1}uI + 12 \ C[42]uu_{4 x}I + -14 C[42]u_{3 x}D_x^{2} + 15 C[42]u_{2 x}^{2}I + -15 \ C[42]u_{2 x}D_x^{3} + 18 C[42]uu_{x}^{2}D_x^{1} + 23 C[42]u_{x}u_{3 x}I + \ 2C[42]u_{5 x}D_x^{-1}uI + 2C[42]u_{x}D_x^{-1}u_{4 x}I + \ 2C[42]u_{x}D_x^{-1}u^{5}I + 30 C[42]uu_{3 x}D_x^{1} + 30 C[42]uu_{x}D_x^{3} + \ 31 C[42]u_{x}^{2}D_x^{2} + -38 C[42]u^{3}u_{2 x}I + 38 C[42]uu_{x}u_{2 x}I + \ 3 C[42]u^{2}u_{3 x}I + 40 C[42]uu_{2 x}D_x^{2} + -40C[42]uu_{x}u_{2 \ x}D_x^{-1}uI + 4 C[42]u^{6}I + -54 C[42]u^{3}u_{x}D_x^{1} + 63 C[42]u_{x}u_{2 \ x}D_x^{1} + 6 C[42]u^{2}D_x^{4} + 6 C[42]u^{2}u_{x}D_x^{2} + -6 C[42]u_{4 \ x}D_x^{1} + 6 C[42]u_{x}^{3}I + -6 C[42]u_{x}D_x^{4} + -74 \ C[42]u^{2}u_{x}^{2}I + 9 C[42]u^{2}u_{2 x}D_x^{1} + -9 C[42]u^{4}D_x^{2} + \ -C[42]D_x^{6} + -C[42]u_{5 x}I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\(-C[42]\)*operatorD[6]\), "+", RowBox[{\((\(-15\)\ C[42])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\((\(-14\)\ C[42])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", \(\((\(-9\)\ C[42])\)*u[x, t]\^4*operatorD[2]\), "+", RowBox[{\((\(-6\)\ C[42])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[4]\)}], "+", RowBox[{\((\(-6\)\ C[42])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((4\ C[42])\)*u[x, t]\^6*operatorD[0]\), "+", \(\((6\ C[42])\)*u[x, t]\^2*operatorD[4]\), "+", RowBox[{\((6\ C[42])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "3"], "*", \(operatorD[0]\)}], "+", RowBox[{\((15\ C[42])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\((31\ C[42])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-74\)\ C[42])\), "*", \(u[x, t]\^2\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-54\)\ C[42])\), "*", \(u[x, t]\^3\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-38\)\ C[42])\), "*", \(u[x, t]\^3\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((3\ C[42])\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((6\ C[42])\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\((9\ C[42])\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((12\ C[42])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((18\ C[42])\), "*", \(u[x, t]\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[1]\)}], "+", RowBox[{\((23\ C[42])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((30\ C[42])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\((30\ C[42])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((40\ C[42])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\((63\ C[42])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((38\ C[42])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-10\), "*", \(C[42]\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "3"], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^5\), "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"10", "*", \(C[42]\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-10\), "*", \(C[42]\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-10\), "*", \(C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\(-10\), "*", \(C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"10", "*", \(C[42]\), "*", \(u[x, t]\^4\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"10", "*", \(C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"10", "*", \(C[42]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-40\), "*", \(C[42]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[BoxData[ \(\(\( (*\ Potential\ Kupershmidt\ Equation\ *) \)\(\[IndentingNewLine]\)\( (*\ Should\ have\ a\ recursion\ operator\ of\ rank\ 6. \ *) \)\(\ \[IndentingNewLine]\)\( (*\ Requires\ that\ weight[u]\ \[Rule] \ 0, \ which\ is\ not\ compatible\ with\ InvariantSymmetries . m . \ *) \)\(\[IndentingNewLine]\)\(\ \)\( (*RecursionOperator[\ \[IndentingNewLine]D[u[x, t], t] \[Equal] D[u[x, t], {x, 5}] + w[1]*5*D[u[x, t], {x, 2}]*D[u[x, t], {x, 3}] - w[2]*5*D[u[x, t], x]^2*D[u[x, t], {x, 3}] - w[2]*5*D[u[x, t], x]*D[u[x, t], {x, 2}]^2 + w[3]*D[u[x, t], x]^5, \[IndentingNewLine]u[x, t], {x, t}]\[IndentingNewLine]*) \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Sawada - Kotera\ Equation\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ 5*u[x, t]^2*D[u[x, t], x] + 5*u[x, t]*D[u[x, t], {x, 3}] + 5*D[u[x, t], x]*D[u[x, t], {x, 2}] + D[u[x, t], {x, 5}], u[x, t], {x, t}, \ Seed\ \[Rule] \ 2]\)\)\)], "Input"], Cell[BoxData[ \("{{{-10 C[17]u_{3 x}D_x^{1} + -11 C[17]u_{2 x}D_x^{2} + -16 C[17]uu_{2 \ x}I + -1C[17]u_{x}D_x^{-1}u^{2}I + -21 C[17]uu_{x}D_x^{1} + \ -2C[17]u_{x}D_x^{-1}u_{2 x}I + -4 C[17]u^{3}I + 5-C[17]u^{2}u_{x}D_x^{-1} + \ -5 C[17]u_{4 x}I + 5-C[17]uu_{3 x}D_x^{-1} + 5-C[17]u_{x}u_{2 x}D_x^{-1} + -6 \ C[17]uD_x^{4} + -6 C[17]u_{x}^{2}I + -9 C[17]u^{2}D_x^{2} + -9 \ C[17]u_{x}D_x^{3} + -C[17]D_x^{6} + -C[17]u_{5 x}D_x^{-1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\(-C[17]\)*operatorD[6]\), "+", RowBox[{\((\(-11\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-10\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((\(-9\)\ C[17])\)*u[x, t]\^2*operatorD[2]\), "+", RowBox[{\((\(-9\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", \(\((\(-6\)\ C[17])\)*u[x, t]*operatorD[4]\), "+", RowBox[{\((\(-6\)\ C[17])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-5\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((\(-4\)\ C[17])\)*u[x, t]\^3*operatorD[0]\), "+", RowBox[{\(-C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\((\(-21\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-16\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"5", "*", \(-C[17]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"5", "*", \(-C[17]\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"5", "*", \(-C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-2\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-1\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Potential\ Sawada - Kotera\ Equation*) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ 5/3*D[u[x, t], x]^3 + 5*D[u[x, t], x]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}], u[x, t], {x, t}, \ Seed\ \[Rule] \ 2\n]\)\)\)], "Input"], Cell[BoxData[ \("{{{-12-C[36]D_x^{-1}u_{2 x}u_{3 x}I + -13 C[36]u_{x}u_{3 x}I + \ -2-C[36]D_x^{-1}u_{6 x}I + -2 C[36]u_{4 x}D_x^{1} + 2C[36]u_{x}D_x^{-1}u_{4 \ x}I + 2C[36]u_{x}D_x^{-1}u_{x}u_{2 x}I + -3 C[36]u_{2 x}^{2}I + -3 C[36]u_{2 \ x}D_x^{3} + -3 C[36]u_{5 x}I + -3 C[36]u_{x}u_{2 x}D_x^{1} + \ -4-C[36]D_x^{-1}u_{x}^{2}u_{2 x}I + -4 C[36]u_{x}^{3}I + \ -6-C[36]D_x^{-1}u_{x}u_{4 x}I + -6 C[36]u_{x}D_x^{4} + -8 C[36]u_{3 x}D_x^{2} \ + -9 C[36]u_{x}^{2}D_x^{2} + -C[36]D_x^{6}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\(-C[36]\)*operatorD[6]\), "+", RowBox[{\((\(-9\)\ C[36])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-8\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-6\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[4]\)}], "+", RowBox[{\((\(-4\)\ C[36])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "3"], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-3\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\((\(-3\)\ C[36])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-3\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-2\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-13\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-3\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-2\), "*", \(-C[36]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((6, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-12\), "*", \(-C[36]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-6\), "*", \(-C[36]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-4\), "*", \(-C[36]\), "*", \(operatorD[\(-1\)]\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[36]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[36]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Kaup - Kupershmidt\ Equation\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ 20*u[x, t]^2*D[u[x, t], x] + 25*D[u[x, t], x]*D[u[x, t], {x, 2}] + 10*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}], u[x, t], {x, t}, \ Seed \[Rule] 2\n]\)\)\)], "Input"], Cell[BoxData[ \("{{{10-2 C[17]uu_{3 x}D_x^{-1} + -120 C[17]uu_{x}D_x^{1} + -12 \ C[17]uD_x^{4} + -13 C[17]u_{4 x}I + 20-2 C[17]u^{2}u_{x}D_x^{-1} + 25-2 \ C[17]u_{x}u_{2 x}D_x^{-1} + -2 C[17]u_{5 x}D_x^{-1} + \ -2C[17]u_{x}D_x^{-1}u_{2 x}I + -32 C[17]u^{3}I + -35 C[17]u_{3 x}D_x^{1} + \ -36 C[17]u^{2}D_x^{2} + -36 C[17]u_{x}D_x^{3} + -49 C[17]u_{2 x}D_x^{2} + -69 \ C[17]u_{x}^{2}I + -82 C[17]uu_{2 x}I + -8C[17]u_{x}D_x^{-1}u^{2}I + \ -C[17]D_x^{6}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\(-C[17]\)*operatorD[6]\), "+", RowBox[{\((\(-69\)\ C[17])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-49\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", \(\((\(-36\)\ C[17])\)*u[x, t]\^2*operatorD[2]\), "+", RowBox[{\((\(-36\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\((\(-35\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((\(-32\)\ C[17])\)*u[x, t]\^3*operatorD[0]\), "+", RowBox[{\((\(-13\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((\(-12\)\ C[17])\)*u[x, t]*operatorD[4]\), "+", RowBox[{\((\(-2\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\((\(-120\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-82\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"10", "*", \((\(-2\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{ "20", "*", \((\(-2\)\ C[17])\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"25", "*", \((\(-2\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-8\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Kaup - Kupershmidt\ Equation\ \((scaled)\)\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ 5*u[x, t]^2*D[u[x, t], x] + 25/2*D[u[x, t], x]*D[u[x, t], {x, 2}] + 5*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}], u[x, t], {x, t}, \ Seed\ \[Rule] \ 2\n]\)\)\)], "Input"], Cell[BoxData[ \("{{{-120 C[17]uu_{x}D_x^{1} + -16 C[17]u^{3}I + -24 C[17]uD_x^{4} + -26 \ C[17]u_{4 x}I + -2C[17]u_{x}D_x^{-1}u_{2 x}I + -36 C[17]u^{2}D_x^{2} + -4 \ C[17]D_x^{6} + -4 C[17]u_{5 x}D_x^{-1} + -4C[17]u_{x}D_x^{-1}u^{2}I + 5-4 \ C[17]u^{2}u_{x}D_x^{-1} + 5-4 C[17]uu_{3 x}D_x^{-1} + -69 C[17]u_{x}^{2}I + \ -70 C[17]u_{3 x}D_x^{1} + -72 C[17]u_{x}D_x^{3} + -82 C[17]uu_{2 x}I + -98 \ C[17]u_{2 x}D_x^{2} + \\frac{25}{2}-4 C[17]u_{x}u_{2 x}D_x^{-1}}}}"\)], \ "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\((\(-4\)\ C[17])\)*operatorD[6]\), "+", RowBox[{\((\(-98\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-72\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\((\(-70\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-69\)\ C[17])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", \(\((\(-36\)\ C[17])\)*u[x, t]\^2*operatorD[2]\), "+", RowBox[{\((\(-26\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((\(-24\)\ C[17])\)*u[x, t]*operatorD[4]\), "+", \(\((\(-16\)\ C[17])\)*u[x, t]\^3*operatorD[0]\), "+", RowBox[{\((\(-4\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\((\(-120\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-82\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"5", "*", \((\(-4\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"5", "*", \((\(-4\)\ C[17])\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(25\/2\), "*", \((\(-4\)\ C[17])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-4\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Potential\ Kaup - Kupershmidt\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ D[u[x, t], \ {x, 5}]\ + \ 10*D[u[x, t], \ x]*D[u[x, t], \ {x, \ 3}]\ + \ 15/2*D[u[x, t], \ {x, \ 2}]^2\ + \ 20/3*D[u[x, t], \ x]^3, \[IndentingNewLine]u[x, t], {x, t}, Seed\ \[Rule] 2\n]\)\)\)], "Input"], Cell[BoxData[ \("{{{-10 C[36]u_{4 x}D_x^{1} + -12 C[36]u_{x}D_x^{4} + \ 16C[36]u_{x}D_x^{-1}u_{x}u_{2 x}I + -21 C[36]u_{2 x}^{2}I + -24 C[36]u_{2 \ x}D_x^{3} + -24\\frac{-1}{2} C[36]D_x^{-1}u_{x}u_{4 x}I + -25 C[36]u_{3 \ x}D_x^{2} + 2C[36]u_{x}D_x^{-1}u_{4 x}I + -2\\frac{-1}{2} C[36]D_x^{-1}u_{6 \ x}I + -32 C[36]u_{x}^{3}I + -34 C[36]u_{x}u_{3 x}I + -36 \ C[36]u_{x}^{2}D_x^{2} + -3 C[36]u_{5 x}I + -48 C[36]u_{x}u_{2 x}D_x^{1} + -48\ \\frac{-1}{2} C[36]D_x^{-1}u_{2 x}u_{3 x}I + -64\\frac{-1}{2} \ C[36]D_x^{-1}u_{x}^{2}u_{2 x}I + -C[36]D_x^{6}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\(-C[36]\)*operatorD[6]\), "+", RowBox[{\((\(-36\)\ C[36])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-32\)\ C[36])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "3"], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-25\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-24\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\((\(-21\)\ C[36])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-12\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[4]\)}], "+", RowBox[{\((\(-10\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-3\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((\(-48\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-34\)\ C[36])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \(-\(C[36]\/2\)\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((6, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-64\), "*", \(-\(C[36]\/2\)\), "*", \(operatorD[\(-1\)]\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-48\), "*", \(-\(C[36]\/2\)\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-24\), "*", \(-\(C[36]\/2\)\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[36]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"16", "*", \(C[36]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Gardner\ Equation\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[w[x, t], \ t]\ == 6*\((w[x, t]\ + \ \[Epsilon]^2*w[x, t]^2)\)*D[w[x, t], \ x]\ - \ D[w[x, t], \ {x, 3}], \[IndentingNewLine]w[x, t], {x, t}, {\[Epsilon]}, Seed \[Rule] 2]\)\)\)], "Input"], Cell[BoxData[ \(weight::"nonuniform1" \(\(:\)\(\ \)\) "Given system has at least one equation with terms of unequal rank, so \ that scaling properties can not be determined."\)], "Message"], Cell[BoxData[ \(weight::"nonuniform2" \(\(:\)\(\ \)\) "Incompatibility has been cured by assuming auxiliary parameters with \ weight."\)], "Message"], Cell[BoxData[ \("{{{2-1w_{x}D_x^{-1} + 2-2 \[Epsilon]^{2}w_{x}D_x^{-1}wI + -4wI + -4 \ \[Epsilon]^{2}w^{2}I + C[1]I + D_x^{2}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(DefiningEquation`Operator`Times[operatorD[2]]\), "+", \(C[1]*operatorD[0]\), "+", \(\(-4\)*w[x, t]*operatorD[0]\), "+", \(\((\(-4\)\ \[Epsilon]\^2)\)*w[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(-1\), "*", RowBox[{ SuperscriptBox["w", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"2", "*", \((\(-2\)\ \[Epsilon]\^2)\), "*", RowBox[{ SuperscriptBox["w", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(w[x, t]\), "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Lax\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[ D[u[x, t], \ t]\ \[Equal] \ 30*u[x, t]^2*D[u[x, t], x] + 20*D[u[x, t], x]*D[u[x, t], {x, 2}] + 10*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}], u[x, t], {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[3]uI + C[3]u_{x}D_x^{-1} + \\frac{1}{2} C[3]D_x^{2}}}}"\)], \ "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(C[3]\/2*operatorD[2]\), "+", RowBox[{\(C[3]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[3])\)*u[x, t]*operatorD[0]\)}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Caudrey - Dodd - Gibbon - Sawada - Kotera\ Equation\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]D[u[x, t], t]\ == \ \(-D[u[x, t], \ {x, \ 5}]\)\ - \ 30*u[x, t]*D[u[x, t], \ {x, 3}]\ - \ 30*D[u[x, t], \ x]*D[u[x, t], \ {x, 2}]\ - \ 180*u[x, t]^2*D[u[x, t], x], \[IndentingNewLine]u[x, t], \[IndentingNewLine]{x, t}, \ Seed \[Rule] 2\[IndentingNewLine]]\)\)\)], "Input"], Cell[BoxData[ \("{{{-16 C[17]uu_{2 x}I + 180\\frac{-1}{6} C[17]u^{2}u_{x}D_x^{-1} + -21 \ C[17]uu_{x}D_x^{1} + -24 C[17]u^{3}I + -2C[17]u_{x}D_x^{-1}u_{2 x}I + \ 30\\frac{-1}{6} C[17]uu_{3 x}D_x^{-1} + 30\\frac{-1}{6} C[17]u_{x}u_{2 \ x}D_x^{-1} + -6 C[17]u_{x}^{2}I + -6C[17]u_{x}D_x^{-1}u^{2}I + -9 \ C[17]u^{2}D_x^{2} + -C[17]uD_x^{4} + \\frac{-11}{6} C[17]u_{2 x}D_x^{2} + \ \\frac{-1}{36} C[17]D_x^{6} + \\frac{-1}{6} C[17]u_{5 x}D_x^{-1} + \ \\frac{-3}{2} C[17]u_{x}D_x^{3} + \\frac{-5}{3} C[17]u_{3 x}D_x^{1} + \ \\frac{-5}{6} C[17]u_{4 x}I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"{", RowBox[{\(\(-\(C[17]\/36\)\)*operatorD[6]\), "+", \(\((\(-24\)\ C[17])\)*u[x, t]\^3*operatorD[0]\), "+", \(\((\(-9\)\ C[17])\)*u[x, t]\^2*operatorD[2]\), "+", RowBox[{\((\(-6\)\ C[17])\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\(-\(\(11\ C[17]\)\/6\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\(-\(\(5\ C[17]\)\/3\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-\(\(3\ C[17]\)\/2\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", \(\(-C[17]\)*u[x, t]*operatorD[4]\), "+", RowBox[{\(-\(\(5\ C[17]\)\/6\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-\(C[17]\/6\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\((\(-21\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-16\)\ C[17])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"30", "*", \(-\(C[17]\/6\)\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"30", "*", \(-\(C[17]\/6\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"180", "*", \(-\(C[17]\/6\)\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-6\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \(C[17]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}]}], "}"}], "}"}], "}"}]], "Output"] }, Open ]], Cell[BoxData[ \(\[IndentingNewLine]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Dispersiveless\ Long\ Wave\ System\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[{D[u[x, t], \ t]\ \[Equal] \ u[x, t]*D[v[x, t], x] + D[u[x, t], x]*v[x, t], D[v[x, t], \ t]\ \[Equal] \ D[u[x, t], x] + v[x, t]*D[v[x, t], x]}, {u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \(weight::"nonuniform3" \(\(:\)\(\ \)\) "The weights, \!\({\(\(\(\(weight[x]\)\) \[Rule] \(\(-1\)\)\)\), \(\(\(\ \(weight[t]\)\) \[Rule] \(\(\(\(-1\)\) - \(\(weight[v]\)\)\)\)\)\), \ \(\(\(\(weight[u]\)\) \[Rule] \(\(2\\ \(\(weight[v]\)\)\)\)\)\)}\), could not \ be fixed. The weights may be fixed using the WeightRules option."\)], \ "Message"], Cell[BoxData[ \($Aborted\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RecursionOperator[{D[u[x, t], \ t]\ \[Equal] \ u[x, t]*D[v[x, t], x] + D[u[x, t], x]*v[x, t], D[v[x, t], \ t]\ \[Equal] \ D[u[x, t], x] + v[x, t]*D[v[x, t], x]}, {u[x, t], v[x, t]}, {x, t}, \ WeightRules\ \[Rule] \ {weight[v] \[Rule] 1}]\)], "Input"], Cell[BoxData[ \("{{{\\frac{1}{2} C[8]vI, C[8]uI + \\frac{1}{2} C[8]u_{x}D_x^{-1}}, \ {C[8]I, \\frac{1}{2} C[8]vI + \\frac{1}{2} C[8]v_{x}D_x^{-1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{\(C[8]\/2*v[x, t]*operatorD[0]\), ",", RowBox[{ RowBox[{\(C[8]\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(C[8]*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{\(C[8]*operatorD[0]\), ",", RowBox[{\(C[8]\/2*v[x, t]*operatorD[0]\), "+", RowBox[{\(C[8]\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Diffusion\ System\ *) \)\(\[IndentingNewLine]\)\( (*\ Using\ RankShift\ \[Rule] \ \(-1\)\ gives\ the\ recursion\ operator\ \ reported\ in\ JP\ Wang' s\ \(\(Thesis\)\(.\)\)\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[{D[u[x, t], \ t]\ \[Equal] \ D[u[x, t], {x, 2}] + v[x, t]^2, D[v[x, t], t]\ \[Equal] \ D[v[x, t], {x, 2}]}, {u[x, t], v[x, t]}, {x, t}, WeightRules \[Rule] {weight[u] -> weight[v]}, RankShift \[Rule] \(-1\)\ ]\)\)\)], "Input"], Cell[BoxData[ \("{{{C[5]D_x^{1}, C[2]D_x^{1} + C[5]vD_x^{-1}}, {0, C[5]D_x^{1}}}}"\)], \ "Print"], Cell[BoxData[ \({{{C[5]*operatorD[1], C[2]*operatorD[1] + C[5]*v[x, t]*operatorD[\(-1\)]}, {0, C[5]*operatorD[1]}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ AKNS\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[{D[u[x, t], \ t]\ \[Equal] \ 2*u[x, t]^2*v[x, t] - D[u[x, t], {x, 2}], D[v[x, t], \ t]\ \[Equal] \ \(-2\)*u[x, t]*v[x, t]^2 + D[v[x, t], {x, 2}]}, {u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \(weight::"nonuniform3" \(\(:\)\(\ \)\) "The weights, \!\({\(\(\(\(weight[x]\)\) \[Rule] \(\(-1\)\)\)\), \(\(\(\ \(weight[t]\)\) \[Rule] \(\(-2\)\)\)\), \(\(\(\(weight[u]\)\) \[Rule] \(\(2 - \ \(\(weight[v]\)\)\)\)\)\)}\), could not be fixed. The weights may be fixed \ using the WeightRules option."\)], "Message"], Cell[BoxData[ \($Aborted\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RecursionOperator[{D[u[x, t], \ t]\ \[Equal] \ 2*u[x, t]^2*v[x, t] - D[u[x, t], {x, 2}], D[v[x, t], \ t]\ \[Equal] \ \(-2\)*u[x, t]*v[x, t]^2 + D[v[x, t], {x, 2}]}, {u[x, t], v[x, t]}, {x, t}, \ WeightRules \[Rule] {weight[u]\ \[Rule] \ weight[v]}]\)], "Input"], Cell[BoxData[ \("{{{-1C[16]uD_x^{-1}vI + \\frac{1}{2} C[16]D_x^{1}, \ -1C[16]uD_x^{-1}uI}, {C[16]vD_x^{-1}vI, C[16]vD_x^{-1}uI + \\frac{-1}{2} \ C[16]D_x^{1}}}}"\)], "Print"], Cell[BoxData[ \({{{C[16]\/2*operatorD[1] + \(-1\)*C[16]*u[x, t]*operatorD[\(-1\)]* v[x, t]*operatorD[0], \(-1\)*C[16]*u[x, t]*operatorD[\(-1\)]* u[x, t]*operatorD[0]}, {C[16]*v[x, t]*operatorD[\(-1\)]*v[x, t]* operatorD[0], \(-\(C[16]\/2\)\)*operatorD[1] + C[16]*v[x, t]*operatorD[\(-1\)]*u[x, t]* operatorD[0]}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Nonlinear\ Schrodinger\ System\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\[IndentingNewLine]{D[ u[x, t], t] \[Equal] D[v[x, t], {x, 2}] + v[x, t]*\((u[x, t]^2 + v[x, t]^2)\), \[IndentingNewLine]D[ v[x, t], t] \[Equal] \(-D[u[x, t], {x, 2}]\) - u[x, t]*\((u[x, t]^2 + v[x, t]^2)\)}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{-2C[16]vD_x^{-1}uI, -2C[16]vD_x^{-1}vI + -C[16]D_x^{1}}, \ {2C[16]uD_x^{-1}uI + C[16]D_x^{1}, 2C[16]uD_x^{-1}vI}}}"\)], "Print"], Cell[BoxData[ \({{{\(-2\)*C[16]*v[x, t]*operatorD[\(-1\)]*u[x, t]* operatorD[0], \(-C[16]\)*operatorD[1] + \(-2\)*C[16]*v[x, t]* operatorD[\(-1\)]*v[x, t]*operatorD[0]}, {C[16]*operatorD[1] + 2*C[16]*u[x, t]*operatorD[\(-1\)]*u[x, t]*operatorD[0], 2*C[16]*u[x, t]*operatorD[\(-1\)]*v[x, t]* operatorD[0]}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(RecursionOperator[\[IndentingNewLine]{D[u[x, t], t] \[Equal] D[v[x, t], {x, 2}] - v[x, t]*\((u[x, t]^2 + v[x, t]^2)\), \[IndentingNewLine]D[ v[x, t], t] \[Equal] \(-D[u[x, t], {x, 2}]\) + u[x, t]*\((u[x, t]^2 + v[x, t]^2)\)}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}]\)], "Input"], Cell[BoxData[ \("{{{-2C[16]vD_x^{-1}uI, -2C[16]vD_x^{-1}vI + C[16]D_x^{1}}, \ {2C[16]uD_x^{-1}uI + -C[16]D_x^{1}, 2C[16]uD_x^{-1}vI}}}"\)], "Print"], Cell[BoxData[ \({{{\(-2\)*C[16]*v[x, t]*operatorD[\(-1\)]*u[x, t]*operatorD[0], C[16]*operatorD[1] + \(-2\)*C[16]*v[x, t]*operatorD[\(-1\)]*v[x, t]* operatorD[0]}, {\(-C[16]\)*operatorD[1] + 2*C[16]*u[x, t]*operatorD[\(-1\)]*u[x, t]*operatorD[0], 2*C[16]*u[x, t]*operatorD[\(-1\)]*v[x, t]* operatorD[0]}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Derivative\ Nonlinear\ Schrodinger\ System\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\[IndentingNewLine]{D[ u[x, t], t] \[Equal] \(-D[v[x, t], {x, 2}]\) - \((u[x, t]^2 + v[x, t]^2)\)* D[u[x, t], x], \[IndentingNewLine]D[v[x, t], t] \[Equal] D[u[x, t], {x, 2}] - \((u[x, t]^2 + v[x, t]^2)\)* D[v[x, t], x]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2C[24]u_{x}D_x^{-1}uI + -2 C[24]vD_x^{-1}v_{x}I + C[24]u^{2}I + \ C[24]v^{2}I, -1-2 C[24]vD_x^{-1}u_{x}I + 2 C[24]D_x^{1} + \ 2C[24]u_{x}D_x^{-1}vI}, {-1-2 C[24]uD_x^{-1}v_{x}I + -2 C[24]D_x^{1} + \ 2C[24]v_{x}D_x^{-1}uI, -2 C[24]uD_x^{-1}u_{x}I + 2C[24]v_{x}D_x^{-1}vI + \ C[24]u^{2}I + C[24]v^{2}I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(C[24]*u[x, t]\^2*operatorD[0]\), "+", \(C[24]*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\((\(-2\)\ C[24])\), "*", \(v[x, t]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[24]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(\((2\ C[24])\)*operatorD[1]\), "+", RowBox[{\(-1\), "*", \((\(-2\)\ C[24])\), "*", \(v[x, t]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[24]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(\((\(-2\)\ C[24])\)*operatorD[1]\), "+", RowBox[{\(-1\), "*", \((\(-2\)\ C[24])\), "*", \(u[x, t]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[24]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(C[24]*u[x, t]\^2*operatorD[0]\), "+", \(C[24]*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\((\(-2\)\ C[24])\), "*", \(u[x, t]\), "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[24]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Modified\ Derivative\ Nonlinear\ Schrodinger\ System\ *) \)\(\ \[IndentingNewLine]\)\(RecursionOperator[\[IndentingNewLine]{D[u[x, t], \ t]\ == \ D[ u[x, t]*\((u[x, t]^2 + v[x, t]^2)\)\ + \ \[Beta]*u[x, t] - D[v[x, t], x], x], \[IndentingNewLine]D[v[x, t], t]\ == \[IndentingNewLine]D[ v[x, t]*\((u[x, t]^2 + v[x, t]^2)\) + D[u[x, t], x], x]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}, {\[Beta]}, WeightedParameters\ \[Rule] \ {\[Beta]}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2\\frac{1}{2} C[15]u_{x}D_x^{-1}uI + C[15]u^{2}I + I, 2\\frac{1}{2} \ C[15]u_{x}D_x^{-1}vI + C[15]uvI + \\frac{-1}{2} C[15]D_x^{1}}, {2\\frac{1}{2} \ C[15]v_{x}D_x^{-1}uI + C[15]uvI + \\frac{1}{2} C[15]D_x^{1}, 2\\frac{1}{2} \ C[15]v_{x}D_x^{-1}vI + C[15]v^{2}I + I + \\frac{-1}{2} \[Beta] C[15]I}}, \ {{2\\frac{1}{2}u_{x}D_x^{-1}uI + C[16]I + \\frac{1}{2} \[Beta]I + u^{2}I, \ 2\\frac{1}{2}u_{x}D_x^{-1}vI + \\frac{-1}{2}D_x^{1} + uvI}, \ {2\\frac{1}{2}v_{x}D_x^{-1}uI + \\frac{1}{2}D_x^{1} + uvI, \ 2\\frac{1}{2}v_{x}D_x^{-1}vI + C[16]I + v^{2}I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(DefiningEquation`Operator`Times[operatorD[0]]\), "+", \(C[15]*u[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(C[15]\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(\(-\(C[15]\/2\)\)*operatorD[1]\), "+", \(C[15]*u[x, t]*v[x, t]*operatorD[0]\), "+", RowBox[{"2", "*", \(C[15]\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(C[15]\/2*operatorD[1]\), "+", \(C[15]*u[x, t]*v[x, t]*operatorD[0]\), "+", RowBox[{"2", "*", \(C[15]\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(DefiningEquation`Operator`Times[operatorD[0]]\), "+", \(\((\(-\(1\/2\)\)\ \[Beta]\ C[15])\)*operatorD[0]\), "+", \(C[15]*v[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(C[15]\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(\[Beta]\/2*operatorD[0]\), "+", \(C[16]*operatorD[0]\), "+", \(u[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(\(-\(1\/2\)\)*operatorD[1]\), "+", \(u[x, t]*v[x, t]*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(1\/2*operatorD[1]\), "+", \(u[x, t]*v[x, t]*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(C[16]*operatorD[0]\), "+", \(v[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}]}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Boussinesq\ System\ [ Ablowitz\ `91]\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]{D[u[x, t], t] \[Equal] D[u[x, t]*v[x, t], x] + D[v[x, t], {x, 3}], \[IndentingNewLine]D[ v[x, t], t] \[Equal] D[u[x, t], x] + v[x, t]*D[v[x, t], x]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{\\frac{1}{2} C[8]vI, C[8]D_x^{2} + C[8]uI + \\frac{1}{2} \ C[8]u_{x}D_x^{-1}}, {C[8]I, \\frac{1}{2} C[8]vI + \\frac{1}{2} \ C[8]v_{x}D_x^{-1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{\(C[8]\/2*v[x, t]*operatorD[0]\), ",", RowBox[{\(C[8]*operatorD[2]\), "+", RowBox[{\(C[8]\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(C[8]*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{\(C[8]*operatorD[0]\), ",", RowBox[{\(C[8]\/2*v[x, t]*operatorD[0]\), "+", RowBox[{\(C[8]\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Boussinesq\ System\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]{D[u[x, t], \ t]\ \[Equal] \ D[v[x, t], x], \[IndentingNewLine]\ D[v[x, t], \ t]\ \[Equal] 1/3*D[u[x, t], \ {x, 3}]\ + 8/3*u[x, t]*D[u[x, t], \ x]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}, \ Seed \[Rule] 2]\)\)\)], "Input"], Cell[BoxData[ \("{{{32 C[7]v_{x}D_x^{-1} + 9 C[7]vI, 3 C[7]D_x^{2} + 3 \ C[7]u_{x}D_x^{-1} + 6 C[7]uI}, {10 C[7]uD_x^{2} + 15 C[7]u_{x}D_x^{1} + 16 \ C[7]u^{2}I + 2 C[7]u_{3 x}D_x^{-1} + 82 C[7]uu_{x}D_x^{-1} + 9 C[7]u_{2 x}I + \ C[7]D_x^{4}, 3 C[7]v_{x}D_x^{-1} + 9 C[7]vI}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(\((9\ C[7])\)*v[x, t]*operatorD[0]\), "+", RowBox[{"3", "*", \((2\ C[7])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], ",", RowBox[{\(\((3\ C[7])\)*operatorD[2]\), "+", RowBox[{\((3\ C[7])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((6\ C[7])\)*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(C[7]*operatorD[4]\), "+", RowBox[{\((2\ C[7])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\((9\ C[7])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((10\ C[7])\)*u[x, t]*operatorD[2]\), "+", RowBox[{\((15\ C[7])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((16\ C[7])\)*u[x, t]\^2*operatorD[0]\), "+", RowBox[{"8", "*", \((2\ C[7])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], ",", RowBox[{ RowBox[{\((3\ C[7])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((9\ C[7])\)*v[x, t]*operatorD[0]\)}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Modified\ Boussinesq\ System\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\[IndentingNewLine]{D[ u[x, t], \ t]\ \[Equal] \ 3*D[v[x, t], {x, 2}] + 6*u[x, t]*D[v[x, t], x] + 6*D[u[x, t], x]*v[x, t], \[IndentingNewLine]D[v[x, t], \ t]\ == \ \(-D[u[x, t], {x, 2}]\) - 6*v[x, t]*D[v[x, t], x] + 2*u[x, t]*D[u[x, t], x]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}, Seed\ \[Rule] \ 2]\)\)\)], "Input"], Cell[BoxData[ \("{{{-12 C[94]u_{x}D_x^{-1}v_{x}I + -22 C[94]u_{x}D_x^{-1}uvI + \ -2C[94]v_{2 x}D_x^{-1}uI + 2 C[94]vD_x^{2} + 3 C[94]v_{x}D_x^{1} + \ -4C[94]uv_{x}D_x^{-1}uI + -4 C[94]uv_{x}I + -4C[94]vu_{x}D_x^{-1}uI + -8 \ C[94]u^{2}vI + C[94]v_{2 x}I, -12C[94]uv_{x}D_x^{-1}vI + -12 \ C[94]u_{x}D_x^{-1}u^{2}I + -12C[94]vu_{x}D_x^{-1}vI + -12 C[94]vv_{x}I + -2 \ C[94]u^{3}I + 2 C[94]uD_x^{2} + 2 C[94]u_{x}D_x^{-1}u_{x}I + 32 \ C[94]u_{x}D_x^{-1}v^{2}I + 3 C[94]u_{x}D_x^{1} + -3 C[94]v^{2}D_x^{1} + -6 \ C[94]uv^{2}I + -6C[94]v_{2 x}D_x^{-1}vI + C[94]D_x^{3} + -C[94]u^{2}D_x^{1} + \ C[94]u_{2 x}I}, {-12 C[94]v_{x}D_x^{-1}v_{x}I + -22 C[94]v_{x}D_x^{-1}uvI + \ -2 C[94]uv^{2}I + 4C[94]vv_{x}D_x^{-1}uI + C[94]u_{x}D_x^{1} + \ C[94]v^{2}D_x^{1} + \\frac{-1}{3} C[94]D_x^{3} + \\frac{1}{3} \ C[94]u^{2}D_x^{1} + \\frac{1}{3} C[94]u_{2 x}I + \\frac{2}{3}C[94]u_{2 \ x}D_x^{-1}uI + \\frac{-2}{3} C[94]u^{3}I + \\frac{2}{3} C[94]uD_x^{2} + \ \\frac{-4}{3}C[94]uu_{x}D_x^{-1}uI + \\frac{4}{3} C[94]uu_{x}I, 12 \ C[94]v^{3}I + 12C[94]vv_{x}D_x^{-1}vI + -12 C[94]v_{x}D_x^{-1}u^{2}I + \ 2C[94]u_{2 x}D_x^{-1}vI + -2 C[94]vD_x^{2} + 2 C[94]v_{x}D_x^{-1}u_{x}I + 32 \ C[94]v_{x}D_x^{-1}v^{2}I + -3 C[94]v_{x}D_x^{1} + -4 C[94]u^{2}vI + \ -4C[94]uu_{x}D_x^{-1}vI + 4 C[94]vu_{x}I + -C[94]v_{2 x}I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\(C[94]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((2\ C[94])\)*v[x, t]*operatorD[2]\), "+", RowBox[{\((3\ C[94])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((\(-8\)\ C[94])\)*u[x, t]\^2*v[x, t]*operatorD[0]\), "+", RowBox[{\((\(-4\)\ C[94])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \(C[94]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-1\), "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-4\), "*", \(C[94]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-4\), "*", \(C[94]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(C[94]*operatorD[3]\), "+", \(\((\(-3\)\ C[94])\)*v[x, t]\^2*operatorD[1]\), "+", \(\((\(-2\)\ C[94])\)*u[x, t]\^3*operatorD[0]\), "+", \(\(-C[94]\)*u[x, t]\^2*operatorD[1]\), "+", RowBox[{\(C[94]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((2\ C[94])\)*u[x, t]*operatorD[2]\), "+", RowBox[{\((3\ C[94])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((\(-12\)\ C[94])\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((\(-6\)\ C[94])\)*u[x, t]*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\((2\ C[94])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-6\), "*", \(C[94]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-1\), "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{"3", "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-12\), "*", \(C[94]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-12\), "*", \(C[94]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(\(-\(C[94]\/3\)\)*operatorD[3]\), "+", \(\(-\(\(2\ C[94]\)\/3\)\)*u[x, t]\^3*operatorD[0]\), "+", \(C[94]\/3*u[x, t]\^2*operatorD[1]\), "+", RowBox[{\(C[94]\/3\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\(2\ C[94]\)\/3*u[x, t]*operatorD[2]\), "+", \(C[94]*v[x, t]\^2*operatorD[1]\), "+", RowBox[{\(C[94]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((\(-2\)\ C[94])\)*u[x, t]*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\(\(4\ C[94]\)\/3\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-1\), "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(2\/3\), "*", \(C[94]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-2\), "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-\(4\/3\)\), "*", \(C[94]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"4", "*", \(C[94]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{ RowBox[{\((\(-3\)\ C[94])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((\(-2\)\ C[94])\)*v[x, t]*operatorD[2]\), "+", RowBox[{\(-C[94]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((12\ C[94])\)*v[x, t]\^3*operatorD[0]\), "+", \(\((\(-4\)\ C[94])\)*u[x, t]\^2*v[x, t]*operatorD[0]\), "+", RowBox[{\((4\ C[94])\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((2\ C[94])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-1\), "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[94]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"3", "*", \((2\ C[94])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-4\), "*", \(C[94]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"12", "*", \(C[94]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Hirota - Satsuma\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]{D[u[x, t], t]\ \[Equal] \ 1/2*\((6*u[x, t]*D[u[x, t], \ x]\ + \ D[u[x, t], \ {x, 3}])\) - 2*v[x, t]*D[v[x, t], \ x], \[IndentingNewLine]D[v[x, t], \ t]\ \[Equal] \ \(-3\)*u[x, t]*D[v[x, t], \ x]\ - \ D[v[x, t], \ {x, 3}]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}, Seed \[Rule] 2]\)\)\)], "Input"], Cell[BoxData[ \("{{{-12 C[49]u^{2}I + 23 C[49]vv_{x}D_x^{-1} + -33 C[49]uu_{x}D_x^{-1} \ + -3C[49]u_{x}D_x^{-1}uI + 4 C[49]v^{2}I + -6 C[49]u_{2 x}I + -6 \ C[49]uD_x^{2} + -9 C[49]u_{x}D_x^{1} + \\frac{-1}{2}3 C[49]u_{3 x}D_x^{-1} + \ \\frac{-3}{4} C[49]D_x^{4}, 2C[49]u_{x}D_x^{-1}vI + 4 C[49]uvI + 4 \ C[49]v_{x}D_x^{1} + 5 C[49]vD_x^{2} + C[49]v_{2 x}I}, {33 C[49]uv_{x}D_x^{-1} \ + 3 C[49]v_{3 x}D_x^{-1} + -3C[49]v_{x}D_x^{-1}uI + 9 C[49]v_{2 x}I + \ \\frac{15}{2} C[49]v_{x}D_x^{1}, 12 C[49]uD_x^{2} + 2C[49]v_{x}D_x^{-1}vI + 3 \ C[49]D_x^{4} + 4 C[49]v^{2}I + 6 C[49]u_{x}D_x^{1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(\(-\(\(3\ C[49]\)\/4\)\)*operatorD[4]\), "+", \(\((\(-12\)\ C[49])\)*u[x, t]\^2*operatorD[0]\), "+", RowBox[{\((\(-9\)\ C[49])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((\(-6\)\ C[49])\)*u[x, t]*operatorD[2]\), "+", RowBox[{\((\(-6\)\ C[49])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((4\ C[49])\)*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\(-\(1\/2\)\), "*", \((3\ C[49])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-3\), "*", \((3\ C[49])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{"2", "*", \((3\ C[49])\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-3\), "*", \(C[49]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{ RowBox[{\(C[49]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\((4\ C[49])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((5\ C[49])\)*v[x, t]*operatorD[2]\), "+", \(\((4\ C[49])\)*u[x, t]*v[x, t]*operatorD[0]\), "+", RowBox[{"2", "*", \(C[49]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{\((3\ C[49])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(\(15\ C[49]\)\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((9\ C[49])\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"3", "*", \((3\ C[49])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-3\), "*", \(C[49]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(\((3\ C[49])\)*operatorD[4]\), "+", \(\((4\ C[49])\)*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\((6\ C[49])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((12\ C[49])\)*u[x, t]*operatorD[2]\), "+", RowBox[{"2", "*", \(C[49]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Symmetrically\ Coupled\ Korteweg - de\ Vries\ System\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]{D[u[x, t], t] \[Equal] D[u[x, t], {x, 3}] + D[v[x, t], {x, 3}] + 6*u[x, t]*D[u[x, t], x] + 4*u[x, t]*D[v[x, t], x] + 2*D[u[x, t], x]*v[x, t], \[IndentingNewLine]D[v[x, t], t] \[Equal] D[u[x, t], {x, 3}] + D[v[x, t], {x, 3}] + 6*v[x, t]*D[v[x, t], x] + 4*v[x, t]*D[u[x, t], x] + 2*D[v[x, t], x]*u[x, t]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}, Seed \[Rule] 2, RankShift \[Rule] \(-1\)]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[16]uI + C[16]u_{x}D_x^{-1} + \\frac{1}{2} C[16]D_x^{2}, 2 \ C[16]uI + C[16]u_{x}D_x^{-1} + \\frac{1}{2} C[16]D_x^{2}}, {2 C[16]vI + \ C[16]v_{x}D_x^{-1} + \\frac{1}{2} C[16]D_x^{2}, 2 C[16]vI + \ C[16]v_{x}D_x^{-1} + \\frac{1}{2} C[16]D_x^{2}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(C[16]\/2*operatorD[2]\), "+", RowBox[{\(C[16]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[16])\)*u[x, t]*operatorD[0]\)}], ",", RowBox[{\(C[16]\/2*operatorD[2]\), "+", RowBox[{\(C[16]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[16])\)*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(C[16]\/2*operatorD[2]\), "+", RowBox[{\(C[16]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[16])\)*v[x, t]*operatorD[0]\)}], ",", RowBox[{\(C[16]\/2*operatorD[2]\), "+", RowBox[{\(C[16]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[16])\)*v[x, t]*operatorD[0]\)}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Complexly\ Coupled\ Korteweg - de\ Vries\ System\ *) \)\(\[IndentingNewLine]\)\( (*\ Generates\ two\ linear\ independent\ recursion\ opeators, \ one\ of\ which\ corresponds\ to\ that\ reported\ in\ JP\ Wang' s\ \(\(thesis\)\(.\)\)\ \ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\[IndentingNewLine]{D[ u[x, t], t] \[Equal] D[u[x, t], {x, 3}] + 6*u[x, t]*D[u[x, t], x] + 6*v[x, t]*D[v[x, t], x], \[IndentingNewLine]D[v[x, t], t] \[Equal] D[v[x, t], {x, 3}] + 6*u[x, t]*D[v[x, t], x] + 6*v[x, t]*D[u[x, t], x]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[19]uI + 2 C[20]vI + C[19]u_{x}D_x^{-1} + C[20]v_{x}D_x^{-1} + \ \\frac{1}{2} C[19]D_x^{2}, 2 C[19]vI + 2 C[20]uI + C[19]v_{x}D_x^{-1} + \ C[20]u_{x}D_x^{-1} + \\frac{1}{2} C[20]D_x^{2}}, {2 C[19]vI + 2 C[20]uI + \ C[19]v_{x}D_x^{-1} + C[20]u_{x}D_x^{-1} + \\frac{1}{2} C[20]D_x^{2}, 2 \ C[19]uI + 2 C[20]vI + C[19]u_{x}D_x^{-1} + C[20]v_{x}D_x^{-1} + \\frac{1}{2} \ C[19]D_x^{2}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(C[19]\/2*operatorD[2]\), "+", RowBox[{\(C[19]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[19])\)*u[x, t]*operatorD[0]\), "+", RowBox[{\(C[20]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[20])\)*v[x, t]*operatorD[0]\)}], ",", RowBox[{\(C[20]\/2*operatorD[2]\), "+", RowBox[{\(C[19]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[19])\)*v[x, t]*operatorD[0]\), "+", RowBox[{\(C[20]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[20])\)*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(C[20]\/2*operatorD[2]\), "+", RowBox[{\(C[19]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[19])\)*v[x, t]*operatorD[0]\), "+", RowBox[{\(C[20]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[20])\)*u[x, t]*operatorD[0]\)}], ",", RowBox[{\(C[19]\/2*operatorD[2]\), "+", RowBox[{\(C[19]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[19])\)*u[x, t]*operatorD[0]\), "+", RowBox[{\(C[20]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[20])\)*v[x, t]*operatorD[0]\)}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Ito\ System\ \((Coupled\ Nonlinear\ Wave\ System)\)\ *) \)\(\ \[IndentingNewLine]\)\(RecursionOperator[\[IndentingNewLine]{D[u[x, t], t] \[Equal] D[u[x, t], {x, 3}] + 6*u[x, t]*D[u[x, t], x] + 2*v[x, t]*D[v[x, t], x], \[IndentingNewLine]D[v[x, t], t] \[Equal] 2*u[x, t]*D[v[x, t], x] + 2*D[u[x, t], x]*v[x, t]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{2 C[15]uI + C[15]u_{x}D_x^{-1} + \\frac{1}{2} C[15]D_x^{2}, \ C[15]vI}, {C[15]vI + C[15]v_{x}D_x^{-1}, 0}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(C[15]\/2*operatorD[2]\), "+", RowBox[{\(C[15]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[15])\)*u[x, t]*operatorD[0]\)}], ",", \(C[15]*v[x, t]*operatorD[0]\)}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(C[15]*v[x, t]*operatorD[0]\), "+", RowBox[{\(C[15]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], ",", "0"}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Drinfel' d - Sokolov - Wilson\ System\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]{D[u[x, t], \ t]\ \[Equal] \ 3*v[x, t]*D[v[x, t], \ x], \[IndentingNewLine]D[v[x, t], \ t]\ \[Equal] \ 2*D[v[x, t], \ {x, 3}]\ + \ 2*u[x, t]*D[v[x, t], \ x]\ + \ D[u[x, t], x]*v[x, t]}, \[IndentingNewLine]{u[x, t], \ v[x, t]}, {x, t}, Seed \[Rule] 3]\)\)\)], "Input"], Cell[BoxData[ \("{{{-2 C[146]u_{x}D_x^{3} + -C[146]u^{2}D_x^{2} + \\frac{-10}{3} \ C[146]uu_{x}D_x^{1} + \\frac{-13}{18} C[146]u_{4 x}I + \\frac{-1}{9} \ C[146]D_x^{6} + \\frac{-1}{9}C[146]u_{5 x}D_x^{-1} + \\frac{-23}{12} \ C[146]u_{x}^{2}I + \\frac{-2}{3} C[146]uD_x^{4} + \\frac{-2}{3}\\frac{-1}{4} \ C[146]u_{x}D_x^{-1}v^{2}I + \\frac{-25}{18}C[146]u_{x}u_{2 x}D_x^{-1} + \ \\frac{25}{3} C[146]vv_{x}D_x^{1} + \\frac{2}{9}\\frac{-1}{4} \ C[146]u_{x}D_x^{-1}u_{2 x}I + \\frac{-35}{18} C[146]u_{3 x}D_x^{1} + \ \\frac{37}{6} C[146]vv_{2 x}I + \\frac{-41}{18} C[146]uu_{2 x}I + \ \\frac{4}{3} C[146]uv^{2}I + \\frac{47}{12} C[146]v_{x}^{2}I + \ \\frac{-49}{18} C[146]u_{2 x}D_x^{2} + \\frac{-4}{9} C[146]u^{3}I + \ \\frac{4}{9}\\frac{-1}{4} C[146]u_{x}D_x^{-1}u^{2}I + \ \\frac{5}{2}C[146]v_{x}v_{2 x}D_x^{-1} + \\frac{5}{3}C[146]uvv_{x}D_x^{-1} + \ \\frac{5}{3}C[146]vv_{3 x}D_x^{-1} + \\frac{5}{6}C[146]v^{2}u_{x}D_x^{-1} + \ \\frac{-5}{9}C[146]u^{2}u_{x}D_x^{-1} + \\frac{-5}{9}C[146]uu_{3 x}D_x^{-1} + \ \\frac{7}{3} C[146]v^{2}D_x^{2}, 2 C[146]v^{3}I + -2\\frac{-1}{4} \ C[146]u_{x}D_x^{-1}v_{2 x}I + 3C[146]vv_{x}D_x^{-1}vI + \\frac{11}{3} \ C[146]uv_{x}D_x^{1} + \\frac{13}{6} C[146]vu_{2 x}I + \\frac{14}{3} \ C[146]vD_x^{4} + \\frac{16}{3} C[146]uvD_x^{2} + \\frac{1}{6} C[146]v_{4 x}I \ + \\frac{17}{3} C[146]v_{x}D_x^{3} + \\frac{20}{3} C[146]vu_{x}D_x^{1} + \ \\frac{2}{3} C[146]u^{2}vI + \\frac{-4}{3}\\frac{-1}{4} \ C[146]u_{x}D_x^{-1}uvI + \\frac{5}{3} C[146]u_{x}v_{x}I + \\frac{5}{6} \ C[146]uv_{2 x}I + \\frac{7}{2} C[146]v_{2 x}D_x^{2} + \\frac{7}{6} C[146]v_{3 \ x}D_x^{1}}, {2 C[146]vu_{x}D_x^{1} + C[146]v_{5 x}D_x^{-1} + \\frac{11}{2} \ C[146]uv_{2 x}I + \\frac{11}{2} C[146]v_{4 x}I + \\frac{14}{9} C[146]vD_x^{4} \ + \\frac{16}{9} C[146]uvD_x^{2} + \\frac{2}{3} C[146]v^{3}I + \ \\frac{-2}{3}\\frac{-1}{4} C[146]v_{x}D_x^{-1}v^{2}I + \\frac{27}{2} \ C[146]v_{2 x}D_x^{2} + \\frac{2}{9} C[146]u^{2}vI + \\frac{2}{9}\\frac{-1}{4} \ C[146]v_{x}D_x^{-1}u_{2 x}I + \\frac{3}{2} C[146]vu_{2 x}I + \ \\frac{35}{18}C[146]v_{x}u_{2 x}D_x^{-1} + \\frac{46}{9} C[146]u_{x}v_{x}I + \ \\frac{4}{9}\\frac{-1}{4} C[146]v_{x}D_x^{-1}u^{2}I + \ \\frac{5}{2}C[146]u_{x}v_{2 x}D_x^{-1} + \\frac{53}{9} C[146]uv_{x}D_x^{1} + \ \\frac{5}{3}C[146]uv_{3 x}D_x^{-1} + \\frac{5}{6}C[146]v^{2}v_{x}D_x^{-1} + \ \\frac{5}{9}C[146]u^{2}v_{x}D_x^{-1} + \\frac{5}{9}C[146]uvu_{x}D_x^{-1} + \ \\frac{5}{9}C[146]vu_{3 x}D_x^{-1} + \\frac{67}{9} C[146]v_{x}D_x^{3} + \ \\frac{73}{6} C[146]v_{3 x}D_x^{1}, 12 C[146]u_{x}D_x^{3} + \ 2C[146]uv_{x}D_x^{-1}vI + 2C[146]v_{3 x}D_x^{-1}vI + -2\\frac{-1}{4} \ C[146]v_{x}D_x^{-1}v_{2 x}I + 3 C[146]D_x^{6} + 3 C[146]u^{2}D_x^{2} + 6 \ C[146]uD_x^{4} + 6 C[146]uu_{x}D_x^{1} + C[146]vu_{x}D_x^{-1}vI + \ \\frac{11}{3} C[146]v^{2}D_x^{2} + \\frac{15}{2} C[146]u_{3 x}D_x^{1} + \ \\frac{27}{2} C[146]u_{2 x}D_x^{2} + \\frac{3}{2} C[146]u_{4 x}I + \ \\frac{3}{2} C[146]uu_{2 x}I + \\frac{3}{4} C[146]u_{x}^{2}I + \\frac{35}{3} \ C[146]vv_{x}D_x^{1} + \\frac{-4}{3}\\frac{-1}{4} C[146]v_{x}D_x^{-1}uvI + \ \\frac{49}{6} C[146]vv_{2 x}I + \\frac{67}{12} C[146]v_{x}^{2}I + \ \\frac{8}{3} C[146]uv^{2}I}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(\(-\(C[146]\/9\)\)*operatorD[6]\), "+", RowBox[{\(-\(\(49\ C[146]\)\/18\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\((\(-2\)\ C[146])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\(-\(\(35\ C[146]\)\/18\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-\(\(23\ C[146]\)\/12\)\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", \(\(-C[146]\)*u[x, t]\^2*operatorD[2]\), "+", RowBox[{\(-\(\(13\ C[146]\)\/18\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\(-\(\(2\ C[146]\)\/3\)\)*u[x, t]*operatorD[4]\), "+", \(\(-\(\(4\ C[146]\)\/9\)\)*u[x, t]\^3*operatorD[0]\), "+", \(\(7\ C[146]\)\/3*v[x, t]\^2*operatorD[2]\), "+", RowBox[{\(\(47\ C[146]\)\/12\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\(-\(1\/9\)\), "*", \(C[146]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-\(\(10\ C[146]\)\/3\)\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-\(\(41\ C[146]\)\/18\)\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\(4\ C[146]\)\/3*u[x, t]*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\(\(37\ C[146]\)\/6\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(25\ C[146]\)\/3\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-\(25\/18\)\), "*", \(C[146]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-\(5\/9\)\), "*", \(C[146]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-\(5\/9\)\), "*", \(C[146]\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(5\/6\), "*", \(C[146]\), "*", \(v[x, t]\^2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(5\/3\), "*", \(C[146]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(5\/2\), "*", \(C[146]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-\(2\/3\)\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(2\/9\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(4\/9\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(5\/3\), "*", \(C[146]\), "*", \(u[x, t]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], ",", RowBox[{ RowBox[{\(C[146]\/6\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(7\ C[146]\)\/6\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\((2\ C[146])\)*v[x, t]\^3*operatorD[0]\), "+", RowBox[{\(\(7\ C[146]\)\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", \(\(14\ C[146]\)\/3*v[x, t]*operatorD[4]\), "+", RowBox[{\(\(17\ C[146]\)\/3\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", \(\(2\ C[146]\)\/3*u[x, t]\^2*v[x, t]*operatorD[0]\), "+", RowBox[{\(\(5\ C[146]\)\/6\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(5\ C[146]\)\/3\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(13\ C[146]\)\/6\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(11\ C[146]\)\/3\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", \(\(16\ C[146]\)\/3*u[x, t]*v[x, t]*operatorD[2]\), "+", RowBox[{\(\(20\ C[146]\)\/3\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-2\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(-\(4\/3\)\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"3", "*", \(C[146]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(\(2\ C[146]\)\/3*v[x, t]\^3*operatorD[0]\), "+", RowBox[{\(C[146]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((5, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\(14\ C[146]\)\/9*v[x, t]*operatorD[4]\), "+", RowBox[{\(\(11\ C[146]\)\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(67\ C[146]\)\/9\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\(\(73\ C[146]\)\/6\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(\(27\ C[146]\)\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", \(\(2\ C[146]\)\/9*u[x, t]\^2*v[x, t]*operatorD[0]\), "+", RowBox[{\(\(3\ C[146]\)\/2\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\(16\ C[146]\)\/9*u[x, t]*v[x, t]*operatorD[2]\), "+", RowBox[{\((2\ C[146])\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(\(46\ C[146]\)\/9\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(11\ C[146]\)\/2\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(53\ C[146]\)\/9\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(5\/9\), "*", \(C[146]\), "*", \(u[x, t]\^2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(5\/9\), "*", \(C[146]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(5\/6\), "*", \(C[146]\), "*", \(v[x, t]\^2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(5\/3\), "*", \(C[146]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(35\/18\), "*", \(C[146]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(5\/2\), "*", \(C[146]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", RowBox[{\(-\(2\/3\)\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(2\/9\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(4\/9\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\^2\), "*", \(operatorD[0]\)}], "+", RowBox[{\(5\/9\), "*", \(C[146]\), "*", \(u[x, t]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}], ",", RowBox[{\(\((3\ C[146])\)*operatorD[6]\), "+", RowBox[{\(\(3\ C[146]\)\/4\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(3\ C[146]\)\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((4, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\((3\ C[146])\)*u[x, t]\^2*operatorD[2]\), "+", \(\(11\ C[146]\)\/3*v[x, t]\^2*operatorD[2]\), "+", RowBox[{\(\(67\ C[146]\)\/12\), "*", SuperscriptBox[ RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "2"], "*", \(operatorD[0]\)}], "+", \(\((6\ C[146])\)*u[x, t]*operatorD[4]\), "+", RowBox[{\(\(15\ C[146]\)\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\((12\ C[146])\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[3]\)}], "+", RowBox[{\(\(27\ C[146]\)\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[2]\)}], "+", RowBox[{\(\(3\ C[146]\)\/2\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", \(\(8\ C[146]\)\/3*u[x, t]*v[x, t]\^2*operatorD[0]\), "+", RowBox[{\((6\ C[146])\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(\(49\ C[146]\)\/6\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{\(\(35\ C[146]\)\/3\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[1]\)}], "+", RowBox[{\(-2\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((2, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[146]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((3, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(C[146]\), "*", \(v[x, t]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{\(-\(4\/3\)\), "*", \(-\(C[146]\/4\)\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}], "+", RowBox[{"2", "*", \(C[146]\), "*", \(u[x, t]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Benney\ System\ *) \)\(\[IndentingNewLine]\)\(RecursionOperator[\ \[IndentingNewLine]{D[u[x, t], t] \[Equal] v[x, t]*D[v[x, t], x] + 2*D[u[x, t]*w[x, t], x], \[IndentingNewLine]D[v[x, t], t] \[Equal] 2*D[u[x, t], x] + D[v[x, t]*w[x, t], x], \[IndentingNewLine]D[ w[x, t], t] \[Equal] 2*D[v[x, t], x] + 2*w[x, t]*D[w[x, t], x]}, \[IndentingNewLine]{u[ x, t], v[x, t], w[x, t]}, {x, t}, WeightRules \[Rule] {weight[w] \[Rule] 1}]\)\)\)], "Input"], Cell[BoxData[ \("{{{C[32]wI, C[32]vI, 2 C[32]uI + C[32]u_{x}D_x^{-1}}, {2 C[32]I, 0, \ C[32]vI + C[32]v_{x}D_x^{-1}}, {0, 2 C[32]I, C[32]wI + C[32]w_{x}D_x^{-1}}}}"\ \)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{\(C[32]*w[x, t]*operatorD[0]\), ",", \(C[32]*v[x, t]*operatorD[0]\), ",", RowBox[{ RowBox[{\(C[32]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[32])\)*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{\(\((2\ C[32])\)*operatorD[0]\), ",", "0", ",", RowBox[{\(C[32]*v[x, t]*operatorD[0]\), "+", RowBox[{\(C[32]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"0", ",", \(\((2\ C[32])\)*operatorD[0]\), ",", RowBox[{\(C[32]*w[x, t]*operatorD[0]\), "+", RowBox[{\(C[32]\), "*", RowBox[{ SuperscriptBox["w", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[BoxData[ \(\(\( (*Dispersive\ Water\ Wave\ System\ *) \)\(\[IndentingNewLine]\)\( \ (*\ Does\ not\ report\ the\ recursion\ operator\ reported\ in\ JP\ Wang' s\ thesis\ *) \)\(\[IndentingNewLine]\)\( (*RecursionOperator[\ \[IndentingNewLine]{D[w[x, t], t] \[Equal] D[v[x, t]*w[x, t], x], D[u[x, t], t] \[Equal] \(-D[u[x, t], {x, 2}]\) + 2*D[u[x, t]*v[x, t], x] + w[x, t]*D[w[x, t], x], \[IndentingNewLine]D[v[x, t], t] \[Equal] D[v[x, t], {x, 2}] - 2*D[u[x, t], x] + 2*v[x, t]*D[v[x, t], x]}, \[IndentingNewLine]{w[x, t], u[x, t], v[x, t]}, {x, t}]\[IndentingNewLine]*) \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ MDV\ System\ *) \)\(\ \)\(\n\)\(RecursionOperator[{D[u[x, t], \ t]\ \[Equal] \ \[Beta]*D[u[x, t], x] + 3*D[u[x, t], x]*u[x, t]^2 + D[u[x, t], x]*v[x, t]^2 + 2*u[x, t]*D[v[x, t], x]*v[x, t] - D[v[x, t], {x, 2}] + \[Gamma]*D[v[x, t], x], D[v[x, t], \ t]\ == D[u[x, t], {x, 2}] + \[Theta]*D[u[x, t], x] + 2*D[u[x, t], x]*u[x, t]*v[x, t] + u[x, t]^2*D[v[x, t], x] + \[Delta]*D[v[x, t], x] + 3*D[v[x, t], x]*v[x, t]^2}, {u[x, t], v[x, t]}, {x, t}, \ {\[Beta], \[Gamma], \[Delta], \[Theta]}, \ WeightedParameters\ \[Rule] \ {\[Beta], \[Gamma], \[Delta], \[Theta]}]\ \)\)\)], "Input"], Cell[BoxData[ \("{{{I, 0}, {0, I}}, {{2\\frac{1}{2}u_{x}D_x^{-1}uI + C[16]I + \ \\frac{1}{2} \[Beta]I + \\frac{-1}{2} \[Delta]I + u^{2}I, \ 2\\frac{1}{2}u_{x}D_x^{-1}vI + \\frac{-1}{2}D_x^{1} + \\frac{1}{2} \[Theta]I \ + uvI}, {2\\frac{1}{2}v_{x}D_x^{-1}uI + \\frac{1}{2}D_x^{1} + \\frac{1}{2} \ \[Theta]I + uvI, 2\\frac{1}{2}v_{x}D_x^{-1}vI + C[16]I + v^{2}I}}}"\)], \ "Print"], Cell[BoxData[ \("{{{C[16]I + \\frac{1}{2}\[Beta]I + \\frac{-1}{2}\[Delta]I + u^{2}I + \ u_{x}D_x^{-1}uI, \\frac{-1}{2}D_x^{1} + \\frac{1}{2}\[Theta]I + uvI + \ u_{x}D_x^{-1}vI}, {\\frac{1}{2}D_x^{1} + \\frac{1}{2}\[Theta]I + uvI + \ v_{x}D_x^{-1}uI, C[16]I + v^{2}I + v_{x}D_x^{-1}vI}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{\({{DefiningEquation`Operator`Times[operatorD[0]], 0}, {0, DefiningEquation`Operator`Times[operatorD[0]]}}\), ",", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{\(\[Beta]\/2*operatorD[0]\), "+", \(\(-\(\[Delta]\/2\)\)*operatorD[0]\), "+", \(C[16]*operatorD[0]\), "+", \(u[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(\(-\(1\/2\)\)*operatorD[1]\), "+", \(\[Theta]\/2*operatorD[0]\), "+", \(u[x, t]*v[x, t]*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{\(1\/2*operatorD[1]\), "+", \(\[Theta]\/2*operatorD[0]\), "+", \(u[x, t]*v[x, t]*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(u[x, t]\), "*", \(operatorD[0]\)}]}], ",", RowBox[{\(C[16]*operatorD[0]\), "+", \(v[x, t]\^2*operatorD[0]\), "+", RowBox[{"2", "*", \(1\/2\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\), "*", \(v[x, t]\), "*", \(operatorD[0]\)}]}]}], "}"}]}], "}"}]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Jaulent\ and\ Miodek\ System\ \((related\ to\ Dispersive\ Water\ \ Wave)\)\ *) \)\(\[IndentingNewLine]\)\( (*\ Clarkson\ and\ Ludlow, \ \[IndentingNewLine]Symmetry\ Reductions, \ Exact\ Solutions, \ and\ Painleve\ Analysis\ for\ a\ Generalized\ Boussinesq\ Equation, \ \ \[IndentingNewLine]J . \ Math . \ Anal . \ and\ \(\(Appl\)\(.\)\), \ 186, \ \(132--\) 155, \ \(\((1994)\)\(.\)\)*) \)\(\[IndentingNewLine]\)\(\ RecursionOperator[\[IndentingNewLine]{D[u[x, t], t] \[Equal] u[x, t]*D[v[x, t], x] + 1/2*D[u[x, t], x]*v[x, t] - 1/4*D[v[x, t], {x, 3}], \[IndentingNewLine]D[v[x, t], t] \[Equal] D[u[x, t], x] + 3/2*v[x, t]*D[v[x, t], x]}, \[IndentingNewLine]{u[ x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{0, 2 C[12]uI + C[12]u_{x}D_x^{-1} + \\frac{-1}{2} C[12]D_x^{2}}, {2 \ C[12]I, 2 C[12]vI + C[12]v_{x}D_x^{-1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{\(\(-\(C[12]\/2\)\)*operatorD[2]\), "+", RowBox[{\(C[12]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[12])\)*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{\(\((2\ C[12])\)*operatorD[0]\), ",", RowBox[{ RowBox[{\(C[12]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[12])\)*v[x, t]*operatorD[0]\)}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\( (*\ Dispersive\ Water\ Wave\ equation\ of\ Kuperschmidt\ *) \)\(\ \[IndentingNewLine]\)\( (*\ Clarkson\ and\ Ludlow, \ \[IndentingNewLine]Symmetry\ Reductions, \ Exact\ Solutions, \ and\ Painleve\ Analysis\ for\ a\ Generalized\ Boussinesq\ Equation, \ \ \[IndentingNewLine]J . \ Math . \ Anal . \ and\ \(\(Appl\)\(.\)\), \ 186, \ \(132--\) 155, \ \(\((1994)\)\(.\)\)*) \)\(\[IndentingNewLine]\)\(\ RecursionOperator[\[IndentingNewLine]{D[u[x, t], t] \[Equal] D[u[x, t]*v[x, t], x] + a*D[v[x, t], {x, 3}] - b*D[u[x, t], {x, 2}], \[IndentingNewLine]D[v[x, t], t] \[Equal] D[u[x, t], x] + v[x, t]*D[v[x, t], x] + b*D[v[x, t], {x, 2}]}\ /. \ {a \[Rule] 1, b \[Rule] 1}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, t}]\)\)\)], "Input"], Cell[BoxData[ \("{{{-2 C[12]D_x^{1} + C[12]vI, 2 C[12]D_x^{2} + 2 C[12]uI + \ C[12]u_{x}D_x^{-1}}, {2 C[12]I, 2 C[12]D_x^{1} + C[12]vI + \ C[12]v_{x}D_x^{-1}}}}"\)], "Print"], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{\(\((\(-2\)\ C[12])\)*operatorD[1] + C[12]*v[x, t]*operatorD[0]\), ",", RowBox[{\(\((2\ C[12])\)*operatorD[2]\), "+", RowBox[{\(C[12]\), "*", RowBox[{ SuperscriptBox["u", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}], "+", \(\((2\ C[12])\)*u[x, t]*operatorD[0]\)}]}], "}"}], ",", RowBox[{"{", RowBox[{\(\((2\ C[12])\)*operatorD[0]\), ",", RowBox[{\(\((2\ C[12])\)*operatorD[1]\), "+", \(C[12]*v[x, t]*operatorD[0]\), "+", RowBox[{\(C[12]\), "*", RowBox[{ SuperscriptBox["v", TagBox[\((1, 0)\), Derivative], MultilineFunction->None], "[", \(x, t\), "]"}], "*", \(operatorD[\(-1\)]\)}]}]}], "}"}]}], "}"}], "}"}]], "Output"] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 951}}, WindowSize->{910, 617}, WindowMargins->{{Automatic, 56}, {Automatic, 32}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 131, 2, 50, "Input"], Cell[1910, 57, 92, 1, 25, "Print"], Cell[2005, 60, 86, 1, 25, "Print"] }, Open ]], Cell[CellGroupData[{ Cell[2128, 66, 260, 5, 90, "Input"], Cell[2391, 73, 83, 1, 25, "Print"], Cell[2477, 76, 500, 13, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[3014, 94, 351, 6, 90, "Input"], Cell[3368, 102, 112, 2, 25, "Print"], Cell[3483, 106, 590, 14, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[4110, 125, 323, 6, 90, "Input"], Cell[4436, 133, 139, 2, 25, "Print"], Cell[4578, 137, 555, 13, 43, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[5170, 155, 344, 6, 70, "Input"], Cell[5517, 163, 200, 3, 37, "Message"], Cell[5720, 168, 160, 3, 22, "Message"], Cell[5883, 173, 73, 1, 25, "Print"], Cell[5959, 176, 454, 11, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[6450, 192, 438, 8, 70, "Input"], Cell[6891, 202, 200, 3, 37, "Message"], Cell[7094, 207, 160, 3, 22, "Message"], Cell[7257, 212, 89, 1, 25, "Print"], Cell[7349, 215, 495, 13, 42, "Output"] }, Open ]], Cell[7859, 231, 514, 8, 130, "Input"], Cell[CellGroupData[{ Cell[8398, 243, 284, 6, 90, "Input"], Cell[8685, 251, 98, 2, 25, "Print"], Cell[8786, 255, 499, 13, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[9322, 273, 263, 6, 70, "Input"], Cell[9588, 281, 92, 1, 25, "Print"], Cell[9683, 284, 750, 18, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[10470, 307, 289, 6, 90, "Input"], Cell[10762, 315, 92, 1, 25, "Print"], Cell[10857, 318, 578, 14, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[11472, 337, 448, 8, 150, "Input"], Cell[11923, 347, 200, 3, 37, "Message"], Cell[12126, 352, 160, 3, 22, "Message"], Cell[12289, 357, 116, 2, 25, "Print"], Cell[12408, 361, 1077, 27, 53, "Output"] }, Open ]], Cell[13500, 391, 533, 10, 130, "Input"], Cell[14036, 403, 458, 7, 130, "Input"], Cell[CellGroupData[{ Cell[14519, 414, 477, 8, 130, "Input"], Cell[14999, 424, 1075, 15, 196, "Print"], Cell[16077, 441, 13515, 315, 432, "Output"] }, Open ]], Cell[29607, 759, 739, 12, 170, "Input"], Cell[CellGroupData[{ Cell[30371, 775, 347, 7, 90, "Input"], Cell[30721, 784, 452, 6, 82, "Print"], Cell[31176, 792, 4765, 111, 166, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[35978, 908, 328, 7, 90, "Input"], Cell[36309, 917, 504, 7, 101, "Print"], Cell[36816, 926, 6908, 170, 205, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[43761, 1101, 351, 7, 110, "Input"], Cell[44115, 1110, 472, 7, 82, "Print"], Cell[44590, 1119, 4824, 112, 167, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[49451, 1236, 369, 7, 110, "Input"], Cell[49823, 1245, 483, 7, 82, "Print"], Cell[50309, 1254, 4823, 111, 206, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[55169, 1370, 416, 9, 130, "Input"], Cell[55588, 1381, 569, 8, 101, "Print"], Cell[56160, 1391, 6960, 171, 255, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[63157, 1567, 345, 6, 90, "Input"], Cell[63505, 1575, 200, 3, 37, "Message"], Cell[63708, 1580, 160, 3, 22, "Message"], Cell[63871, 1585, 146, 2, 25, "Print"], Cell[64020, 1589, 1013, 23, 48, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[65070, 1617, 297, 5, 90, "Input"], Cell[65370, 1624, 98, 2, 25, "Print"], Cell[65471, 1628, 499, 13, 42, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[66007, 1646, 492, 9, 150, "Input"], Cell[66502, 1657, 568, 8, 101, "Print"], Cell[67073, 1667, 4827, 112, 256, "Output"] }, Open ]], Cell[71915, 1782, 52, 1, 50, "Input"], Cell[CellGroupData[{ Cell[71992, 1787, 352, 7, 70, "Input"], Cell[72347, 1796, 371, 6, 37, "Message"], Cell[72721, 1804, 42, 1, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[72800, 1810, 313, 5, 50, "Input"], Cell[73116, 1817, 166, 2, 44, "Print"], Cell[73285, 1821, 1041, 25, 76, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[74363, 1851, 535, 9, 90, "Input"], Cell[74901, 1862, 101, 2, 25, "Print"], Cell[75005, 1866, 155, 3, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[75197, 1874, 318, 5, 70, "Input"], Cell[75518, 1881, 341, 5, 37, "Message"], Cell[75862, 1888, 42, 1, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[75941, 1894, 324, 5, 70, "Input"], Cell[76268, 1901, 178, 3, 44, "Print"], Cell[76449, 1906, 396, 6, 100, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[76882, 1917, 462, 9, 110, "Input"], Cell[77347, 1928, 152, 2, 25, "Print"], Cell[77502, 1932, 386, 6, 86, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[77925, 1943, 365, 6, 90, "Input"], Cell[78293, 1951, 152, 2, 25, "Print"], Cell[78448, 1955, 384, 6, 86, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[78869, 1966, 488, 9, 110, "Input"], Cell[79360, 1977, 340, 5, 63, "Print"], Cell[79703, 1984, 3759, 84, 162, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[83499, 2073, 584, 10, 130, "Input"], Cell[84086, 2085, 588, 8, 101, "Print"], Cell[84677, 2095, 4833, 104, 373, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[89547, 2204, 412, 8, 110, "Input"], Cell[89962, 2214, 182, 3, 44, "Print"], Cell[90147, 2219, 1069, 25, 76, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[91253, 2249, 411, 8, 110, "Input"], Cell[91667, 2259, 294, 4, 63, "Print"], Cell[91964, 2265, 2910, 65, 124, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[94911, 2335, 514, 9, 110, "Input"], Cell[95428, 2346, 1312, 18, 234, "Print"], Cell[96743, 2366, 14844, 314, 646, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[111624, 2685, 489, 8, 110, "Input"], Cell[112116, 2695, 607, 8, 101, "Print"], Cell[112726, 2705, 6311, 137, 254, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[119074, 2847, 681, 13, 150, "Input"], Cell[119758, 2862, 287, 4, 63, "Print"], Cell[120048, 2868, 1945, 45, 144, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[122030, 2918, 726, 14, 150, "Input"], Cell[122759, 2934, 423, 6, 82, "Print"], Cell[123185, 2942, 3445, 77, 231, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[126667, 3024, 494, 10, 110, "Input"], Cell[127164, 3036, 145, 2, 25, "Print"], Cell[127312, 3040, 1072, 26, 67, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[128421, 3071, 481, 9, 110, "Input"], Cell[128905, 3082, 3151, 44, 557, "Print"], Cell[132059, 3128, 27661, 586, 1436, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[159757, 3719, 568, 11, 130, "Input"], Cell[160328, 3732, 183, 3, 44, "Print"], Cell[160514, 3737, 1615, 38, 86, "Output"] }, Open ]], Cell[162144, 3778, 699, 11, 170, "Input"], Cell[CellGroupData[{ Cell[162868, 3793, 743, 14, 150, "Input"], Cell[163614, 3809, 382, 6, 63, "Print"], Cell[163999, 3817, 304, 4, 63, "Print"], Cell[164306, 3823, 2694, 58, 203, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[167037, 3886, 820, 14, 170, "Input"], Cell[167860, 3902, 149, 2, 25, "Print"], Cell[168012, 3906, 1106, 27, 67, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[169155, 3938, 875, 16, 170, "Input"], Cell[170033, 3956, 178, 3, 44, "Print"], Cell[170214, 3961, 1237, 30, 67, "Output"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)