(************** Content-type: application/mathematica ************** 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). 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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[ 56309, 1359]*) (*NotebookOutlinePosition[ 56953, 1381]*) (* CellTagsIndexPosition[ 56909, 1377]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ \(SetDirectory["\"]\)], "Input"], Cell[BoxData[ \(<< PDESpecialSolutions.m\)], "Input"], Cell[BoxData[ \( (*\ 2. \ \ The\ Boussinesq\ System\ *) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[{D[u[x, t], t] + D[v[x, t], x]\ \[Equal] \ 0, \ D[v[x, t], \ t] + D[u[x, t], \ x] - 3*u[x, t]*D[u[x, t], x] - alpha*D[u[x, t], {x, 3}]\ \[Equal] \ 0}, \ {u[x, t], v[x, t]}, \ {x, t}, \ {alpha}, \ Verbose 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To test the \ solutions set either the NumericalTest option to True, or set the \ SymbolicTest option to True, or both. \ \>", "Message"], Cell[BoxData[ \({{{u[x, y, z, t] \[Rule] \(-\@c[2]\)\ \@c[3]\ Sech[\(phase\ c[2]\ c[3] + y\ \ c[2]\^2\ c[3] + z\ c[2]\ c[3]\^2 - x\ c[4] + t\ c[2]\ c[3]\ c[4]\)\/\(c[2]\ \ c[3]\)]}, {u[x, y, z, t] \[Rule] \@c[2]\ \@c[3]\ Sech[\(phase\ c[2]\ c[3] + y\ \ c[2]\^2\ c[3] + z\ c[2]\ c[3]\^2 - x\ c[4] + t\ c[2]\ c[3]\ c[4]\)\/\(c[2]\ \ c[3]\)]}}}\)], "Output"] }, Open ]], Cell[BoxData[ \( (*\ 4. \ \ Gao\ and\ Tian\ *) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[{D[u[x, t], \ t]\ - \ D[u[x, t], \ x]\ - \ 2*v[x, t]\ \[Equal] \ 0, \ D[v[x, t], \ t]\ + \ 2*u[x, t]*w[x, t]\ \[Equal] \ 0, \ D[w[x, t], \ t]\ + \ 2*u[x, t]*v[x, t]\ \[Equal] \ 0}, \ {u[x, t], \ v[x, t], \ w[x, t]}, \ {x, t}, \ {}, \ Form \[Rule] SechTanh]\)], "Input"], Cell["\<\ These solutions are not being tested numerically or symbolically. To test the \ solutions set either the NumericalTest option to True, or set the \ SymbolicTest option to True, or both. \ \>", "Message"], Cell["\<\ The following simplification rules are being used: {Sqrt[a^2]->a, \ Sqrt[-a^2]->I*a}\ \>", "Message"], Cell[BoxData[ \({{{u[x, t] \[Rule] \(-\(\(\@c[2]\ \@\(\((c[1] - c[2])\)\ c[2]\)\ Sech[ phase + x\ c[1] + t\ c[2]]\)\/\@\(\(-c[1]\) + c[2]\)\)\), v[x, t] \[Rule] 1\/2\ \@c[2]\ \@\(\((c[1] - c[2])\)\ c[2]\)\ \@\(\(-c[1]\) + c[2]\ \)\ Sech[phase + x\ c[1] + t\ c[2]]\ Tanh[phase + x\ c[1] + t\ c[2]], w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[ 2]\ \((\(-1\) + 2\ Sech[phase + x\ c[1] + t\ c[2]]\^2)\)}, {u[ x, t] \[Rule] \(\@c[2]\ \@\(\((c[1] - c[2])\)\ c[2]\)\ \ Sech[phase + x\ c[1] + t\ c[2]]\)\/\@\(\(-c[1]\) + c[2]\), v[x, t] \[Rule] \(-\(1\/2\)\)\ \@c[2]\ \@\(\((c[1] - c[2])\)\ \ c[2]\)\ \@\(\(-c[1]\) + c[2]\)\ Sech[phase + x\ c[1] + t\ c[2]]\ Tanh[ phase + x\ c[1] + t\ c[2]], w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[ 2]\ \((\(-1\) + 2\ Sech[phase + x\ c[1] + t\ c[2]]\^2)\)}, {u[ x, t] \[Rule] \(-\(\(\((c[1] - c[2])\)\ c[ 2]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \@\(-\((c[1] - c[2])\)\^2\)\ \)\)\), v[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(\((c[1] - c[2])\)\ c[2]\ \((Sech[phase + x\ c[1] + \ t\ c[2]] - \[ImaginaryI]\ Tanh[phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \ \@\(-\((c[1] - c[2])\)\^2\)\), v[x, t] \[Rule] 1\/4\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] - \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(-\(\(\((c[1] - c[2])\)\ c[ 2]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \@\(-\((c[1] - c[2])\)\^2\)\ \)\)\), v[x, t] \[Rule] 1\/4\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(\((c[1] - c[2])\)\ c[2]\ \((Sech[phase + x\ c[1] + \ t\ c[2]] + \[ImaginaryI]\ Tanh[phase + x\ c[1] + t\ c[2]])\)\)\/\(2\ \ \@\(-\((c[1] - c[2])\)\^2\)\), v[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\), w[x, t] \[Rule] \(-\(1\/4\)\)\ \((c[1] - c[2])\)\ c[2]\ Sech[ phase + x\ c[1] + t\ c[2]]\ \((Sech[ phase + x\ c[1] + t\ c[2]] + \[ImaginaryI]\ Tanh[ phase + x\ c[1] + t\ c[2]])\)}, {u[x, t] \[Rule] \(-\(\(\@c[2]\ \@\(\((c[1] - c[2])\)\^2\ c[2]\)\ \ Tanh[phase + x\ c[1] + t\ c[2]]\)\/\(c[1] - c[2]\)\)\), v[x, t] \[Rule] 1\/2\ \@c[2]\ \@\(\((c[1] - c[2])\)\^2\ c[2]\)\ Sech[phase + x\ \ c[1] + t\ c[2]]\^2, w[x, t] \[Rule] \(-\(1\/2\)\)\ \((c[1] - c[2])\)\ c[ 2]\ Sech[phase + x\ c[1] + t\ c[2]]\^2}, {u[x, t] \[Rule] \(\@c[2]\ \@\(\((c[1] - c[2])\)\^2\ c[2]\)\ \ Tanh[phase + x\ c[1] + t\ c[2]]\)\/\(c[1] - c[2]\), v[x, t] \[Rule] \(-\(1\/2\)\)\ \@c[2]\ \@\(\((c[1] - c[2])\)\^2\ \ c[2]\)\ Sech[phase + x\ c[1] + t\ c[2]]\^2, w[x, t] \[Rule] \(-\(1\/2\)\)\ \((c[1] - c[2])\)\ c[ 2]\ Sech[phase + x\ c[1] + t\ c[2]]\^2}, {u[x, t] \[Rule] a[1, 0] + a[1, 1]\ Sech[phase + t\ c[2] + x\ c[2]] + a[1, 2]\ Sech[phase + t\ c[2] + x\ c[2]]\^2 + b[1, 0]\ Tanh[phase + t\ c[2] + x\ c[2]] + b[1, 1]\ Sech[phase + t\ c[2] + x\ c[2]]\ Tanh[ phase + t\ c[2] + x\ c[2]], v[x, t] \[Rule] 0, w[x, t] \[Rule] 0}}}\)], "Output"] }, Open ]], Cell[BoxData[ \( (*\ 6.1\ \ Zakharov - Kuznetsov\ KdV - type\ equations\ *) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[ D[u[x, y, z, t], t]\ + \ \[IndentingNewLine]alpha*u[x, y, z, t]* D[u[x, y, z, t], x]\ + \[IndentingNewLine]D[ u[x, y, z, t], \ {x, 3}] + \[IndentingNewLine]D[u[x, y, z, t], \ y, \ y, y]\ + \[IndentingNewLine]D[u[x, y, z, t], \ z, \ z, \ z]\ \[Equal] 0, \[IndentingNewLine]u[x, y, z, t], \[IndentingNewLine]{x, y, z, t}, \[IndentingNewLine]{alpha}]\)], "Input"], Cell["\<\ These solutions are not being tested numerically or symbolically. To test the \ solutions set either the NumericalTest option to True, or set the \ SymbolicTest option to True, or both. \ \>", "Message"], Cell[BoxData[ \({{{u[x, y, z, t] \[Rule] \(-\(\(1\/\(alpha\ c[1]\)\)\((\(-8\)\ c[1]\^3 - 8\ c[2]\^3 - 8\ c[3]\^3 + c[4] + 12\ c[1]\^3\ Tanh[phase + x\ c[1] + y\ c[2] + z\ c[3] + t\ \ c[4]]\^2 + 12\ c[2]\^3\ Tanh[phase + x\ c[1] + y\ c[2] + z\ c[3] + t\ \ c[4]]\^2 + 12\ c[3]\^3\ Tanh[phase + x\ c[1] + y\ c[2] + z\ c[3] + t\ \ c[4]]\^2)\)\)\)}}}\)], "Output"] }, Open ]], Cell[BoxData[ \(PDESpecialSolutions[ D[u[x, y, z, t], t]\ + \ \[IndentingNewLine]alpha*u[x, y, z, t]* D[u[x, y, z, t], x]\ + \[IndentingNewLine]D[ u[x, y, z, t], \ {x, 3}] + \[IndentingNewLine]D[u[x, y, z, t], \ x, \ y, y]\ + \[IndentingNewLine]D[u[x, y, z, t], \ x, \ z, \ z]\ \[Equal] 0, \[IndentingNewLine]u[x, y, z, t], \[IndentingNewLine]{x, y, z, t}, \[IndentingNewLine]{alpha}, \ Form \[Rule] Sech]\)], "Input"], Cell[BoxData[ \( (*\ Modified\ KdV - \(ZK\ --\)\ Das\ and\ Verheest\ *) \)], "Input"], Cell[BoxData[ \(PDESpecialSolutions[ D[u[x, y, z, t], \ t]\ + \[IndentingNewLine]alpha\ *\ u[x, y, z, t]\ ^\ 2\ *\ D[u[x, y, z, t], x]\ + \ \[IndentingNewLine]D[ u[x, y, z, t], \ {x, 3}] + \[IndentingNewLine]D[u[x, y, z, t], \ x, \ y, y]\ + \[IndentingNewLine]D[u[x, y, z, t], \ x, \ z, \ z]\ \[Equal] 0, \[IndentingNewLine]u[x, y, z, t], \[IndentingNewLine]{x, y, z, t}, \[IndentingNewLine]{alpha}]\)], "Input"], Cell[BoxData[ \(PDESpecialSolutions[ D[u[x, y, z, t], \ t]\ + \[IndentingNewLine]alpha\ *\ u[x, y, z, t]\ ^\ 2\ *\ D[u[x, y, z, t], x]\ + \ \[IndentingNewLine]D[ u[x, y, z, t], \ {x, 3}] + \[IndentingNewLine]D[u[x, y, z, t], \ x, \ y, y]\ + \[IndentingNewLine]D[u[x, y, z, t], \ x, \ z, \ z]\ \[Equal] 0, \[IndentingNewLine]u[x, y, z, t], \[IndentingNewLine]{x, y, z, t}, \[IndentingNewLine]{alpha}, \ Form \[Rule] Sech]\)], "Input"], Cell[BoxData[ \( (*\ 6.2\ Generalized\ Kuramoto - Sivashinsky\ Equation\ *) \)], "Input"], Cell[BoxData[ \(\(\(\ \)\(PDESpecialSolutions[\[IndentingNewLine]D[u[x, t], t]\ + \ \[IndentingNewLine]u[x, t]*\ D[u[x, t], x]\ + \ \[IndentingNewLine]D[ u[x, t], {x, 2}] + \[IndentingNewLine]alpha*\ D[u[x, t], {x, 3}]\ + \ \[IndentingNewLine]D[ u[x, t], {x, 4}]\ \[Equal] \ 0, \[IndentingNewLine]u[x, t], \[IndentingNewLine]{x, t}, \[IndentingNewLine]{alpha}, \ Verbose \[Rule] True]\)\)\)], "Input"], Cell[BoxData[ \( (*\ 6.3\ \ Coupled\ KdV\ Equations\ *) \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(PDESpecialSolutions[{\[IndentingNewLine]D[u[x, t], t]\ - \[IndentingNewLine]alpha\ *\ \((6\ *\ u[x, t]\ *\ D[u[x, t], x]\ \ + \ \ D[ u[x, t], \ {x, 3}])\)\ + \[IndentingNewLine]2\ *\ beta\ *\ v[x, t]*\ D[v[x, t], \ x]\ \ \[Equal] 0, \[IndentingNewLine]D[v[x, t], t]\ + \ \[IndentingNewLine]3* u[x, t]*D[v[x, t], x]\ + \[IndentingNewLine]D[ v[x, t], {x, 3}]\ \[Equal] \ 0\[IndentingNewLine]}, \[IndentingNewLine]{u[x, t], \ v[x, t]}, {x, t}, {alpha, \ beta}, \ Form \[Rule] Cn]\)], "Input"], Cell[BoxData[ \(PDESpecialSolutionsmSolver::"freedom" \(\(:\)\(\ \)\) "Freedom exists in this system, as \!\({\(\(1 + \(\(m[1]\)\)\)\), \ \(\(\(\(-1\)\) + \(\(2\\ \(\(m[1]\)\)\)\)\)\)}\) are both dominant powers in \ expression \!\(1\)."\)], "Message", GeneratedCell->False, CellAutoOverwrite->False], Cell["\<\ These solutions are not being tested numerically or symbolically. To test the \ solutions set either the NumericalTest option to True, or set the \ SymbolicTest option to True, or both. \ \>", "Message"], Cell[BoxData[ \({{{u[x, t] \[Rule] \(c[1]\^3 - 2\ mod\ c[1]\^3 - c[2] + 6\ mod\ c[1]\^3\ \ JacobiCN[phase + x\ c[1] + t\ c[2], mod]\^2\)\/\(3\ c[1]\), v[x, t] \[Rule] \(-\(\(\@2\ \@mod\ \@c[1]\ \@\(\(-2\)\ alpha\ \ c[1]\^3 + 4\ alpha\ mod\ c[1]\^3 - c[2] - 2\ alpha\ c[2]\)\ JacobiCN[ phase + x\ c[1] + t\ c[2], mod]\)\/\@beta\)\)}, {u[x, t] \[Rule] \(c[1]\^3 - 2\ mod\ c[1]\^3 - c[2] + 6\ mod\ c[1]\^3\ \ JacobiCN[phase + x\ c[1] + t\ c[2], mod]\^2\)\/\(3\ c[1]\), v[x, t] \[Rule] \(\@2\ \@mod\ \@c[1]\ \@\(\(-2\)\ alpha\ c[1]\^3 + \ 4\ alpha\ mod\ c[1]\^3 - c[2] - 2\ alpha\ c[2]\)\ JacobiCN[phase + x\ c[1] + \ t\ c[2], mod]\)\/\@beta}, {u[x, t] \[Rule] \(4\ c[1]\^3 - 8\ mod\ c[1]\^3 - c[2] + 12\ mod\ \ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ c[2], mod]\^2\)\/\(3\ c[1]\), v[x, t] \[Rule] \(-\(\(4\ alpha\ c[1]\^3 - 8\ alpha\ mod\ c[1]\^3 - c[2] - 2\ alpha\ c[2] + 12\ alpha\ mod\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ \ c[2], mod]\^2\)\/\(\@6\ \@alpha\ \@beta\ c[1]\)\)\)}, {u[x, t] \[Rule] \(4\ c[1]\^3 - 8\ mod\ c[1]\^3 - c[2] + 12\ mod\ \ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ c[2], mod]\^2\)\/\(3\ c[1]\), v[x, t] \[Rule] \(4\ alpha\ c[1]\^3 - 8\ alpha\ mod\ c[1]\^3 - c[2] \ - 2\ alpha\ c[2] + 12\ alpha\ mod\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ \ c[2], mod]\^2\)\/\(\@6\ \@alpha\ \@beta\ c[1]\)}, {u[x, t] \[Rule] \(4\ alpha\ c[1]\^3 - 8\ alpha\ mod\ c[1]\^3 + c[2] \ + 12\ alpha\ mod\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ c[2], mod]\^2\)\/\(6\ \ alpha\ c[1]\), v[x, t] \[Rule] 0}, {u[x, t] \[Rule] \(4\ alpha\ c[1]\^3 - 8\ alpha\ mod\ c[1]\^3 + c[2] \ + 12\ alpha\ mod\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ c[2], mod]\^2\)\/\(6\ \ alpha\ c[1]\), v[x, t] \[Rule] a[2, 0]}, {mod \[Rule] \(2\ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ \ c[2]\)\/\(4\ alpha\ c[1]\^3\), u[x, t] \[Rule] \(1\/\(3\ alpha\ c[1]\)\) \((10\ c[2] + 19\ alpha\ c[2] + 6\ alpha\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ \ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2 - 15\ c[2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ alpha\ \ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2 - 30\ alpha\ c[ 2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ alpha\ \ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2)\), v[x, t] \[Rule] \(-\(\(1\/\(\@alpha\ \@beta\ c[ 1]\)\) \((\@\(3\/2\)\ \((3\ c[2] + 6\ alpha\ c[2] + 2\ alpha\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ \ c[2], \(2\ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ \ c[1]\^3\)]\^2 - 5\ c[2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ \ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2 - 10\ alpha\ c[ 2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ \ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2)\))\)\ \)\)}, {mod \[Rule] \(2\ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ \ alpha\ c[1]\^3\), u[x, t] \[Rule] \(1\/\(3\ alpha\ c[1]\)\) \((10\ c[2] + 19\ alpha\ c[2] + 6\ alpha\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ \ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2 - 15\ c[2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ alpha\ \ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2 - 30\ alpha\ c[ 2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ alpha\ \ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2)\), v[x, t] \[Rule] \(1\/\(\@alpha\ \@beta\ c[ 1]\)\) \((\@\(3\/2\)\ \((3\ c[2] + 6\ alpha\ c[2] + 2\ alpha\ c[1]\^3\ JacobiCN[phase + x\ c[1] + t\ c[2], \ \(2\ alpha\ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2 \ - 5\ c[2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ alpha\ c[1]\^3 - 5\ c[2] \ - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2 - 10\ alpha\ c[ 2]\ JacobiCN[phase + x\ c[1] + t\ c[2], \(2\ alpha\ \ c[1]\^3 - 5\ c[2] - 10\ alpha\ c[2]\)\/\(4\ alpha\ c[1]\^3\)]\^2)\))\)}, \ {alpha \[Rule] 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