(************** Content-type: application/mathematica **************

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(*NotebookOptionsPosition[     21255,        527]*)
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Notebook[{
Cell[BoxData[
    \(SetDirectory["\<d:\\yourdirectory\>"]\)], "Input"],

Cell[BoxData[
    \( (*\ 
      yourdirectory\ should\ be\ the\ directory\ where\ the\ code\ and\ \
notebooks\ \(\(reside\)\(.\)\)\ *) \)], "Input"],

Cell[BoxData[
    \(<< PDESpecialSolutions.m\)], "Input"],

Cell[BoxData[
    \(\(\( (*\ 
      Be\ \(aware : \ sometimes\ you\ have\ to\ retype\ the\ << \ above\), \ 
      due\ to\ a\ small\ parsing\ error\ *) \)\(\[IndentingNewLine]\)\( (*\ 
      in\ Mathematica\ *) \)\(\[IndentingNewLine]\)\( (*\ 
      Never\ read\ in\ codes\ twice . \ 
          Quit\ the\ kernel\ prior\ to\ any\ new\ \(\(attempt\)\(.\)\)\ *) \)\
\)\)], "Input"],

Cell[CellGroupData[{

Cell["Standard Cases", "Subtitle"],

Cell[BoxData[
    \( (*\ KdV\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], \ t]\  + \ 6*alpha*u[x, t]*D[u[x, t], x]\  + \ 
          D[u[x, t], {x, 3}]\  \[Equal] \ 0, \ 
      u[x, t], \ {x, t}, \ {alpha}, \ Form \[Rule] Tanh, \ 
      Verbose\  \[Rule] \ True, \ SymbolicTest\  \[Rule] \ True, \ 
      NumericTest\  \[Rule] \ True, \ 
      HighestOrderFirst\  \[Rule] \ False]\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\( (*\ 
      Boussinesq\ Equation\ \((single\ equation)\)\ *) \)\)\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] + 
            3*u[x, t]*D[u[x, t], {x, 2}] + 3*\((D[u[x, t], x])\)^2 + 
            alpha*D[u[x, t], {x, 4}]\  \[Equal] \ 0}, \ {u[x, t]}, \ {x, 
        t}, \ {alpha}, \ Verbose \[Rule] True, \ Form \[Rule] Tanh, \ 
      SymbolicTest \[Rule] True, \ NumericTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\( (*\ 
      2. \ \ The\ Boussinesq\ System\ *) \)\)\)], "Input"],

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 \[Rule] True, \ 
      Form \[Rule] Tanh, \ SymbolicTest \[Rule] True]\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\( (*\ 
      6.3\ \ Coupled\ KdV\ Equations\ *) \)\)\)], "Input"],

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] Sech, \ 
      Verbose\  \[Rule] \ True, \ NumericTest \[Rule] True, \ 
      SymbolicTest\  \[Rule] \ True, \ 
      HighestOrderFirst\  \[Rule] \ False]\)], "Input"],

Cell[BoxData[
    \( (*\ 3. \ \ Modified\ 3  D\ KdV\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, y, z, t], \ t]\  + \ 6*u[x, y, z, t]^2*D[u[x, y, z, t], x]\  + \ 
          D[u[x, y, z, t], {x, 1}, {y, 1}, {z, 1}]\  \[Equal] \ 0, \ 
      u[x, y, z, t], \ {x, y, z, t}, \ {}, \ Form \[Rule] Sech, \ 
      Verbose \[Rule] True]\)], "Input"],

Cell[BoxData[
    \( (*\ 4. \ \ Gao\ and\ Tian\ system\ \ *) \)], "Input"],

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[BoxData[
    \( (*\ 6.1\ \ Zakharov - Kuznetsov\ KdV - type\ equations\ *) \)], "Input"],

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[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\ \ Equation\ *) \)], \
"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}, \ \ Form\  \[Rule] \ 
        Tanh]\)\)\)], "Input"],

Cell[BoxData[
    \( (*\ 6.3\ \ Coupled\ KdV\ Equations\ *) \)], "Input"],

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, \ NumericTest \[Rule] True, \ 
      Verbose \[Rule] True]\)], "Input"],

Cell[BoxData[
    \( (*\ Another\ coupled\ \(KdV\ --\)\ due\ to\ Guha - Roy\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{\[IndentingNewLine]D[u[x, t], 
              t]\  + \ \[IndentingNewLine]\ \[Alpha]\ *\ v[x, t]*
              D[v[x, t], x]\  + \ \[IndentingNewLine]\[Beta]*u[x, t]*
              D[u[x, t], x]\  + \[IndentingNewLine]\[Gamma]\ *\ 
              D[u[x, t], {x, 3}]\  \[Equal] 
          0, \[IndentingNewLine]D[v[x, t], 
              t]\  + \ \[IndentingNewLine]\[Delta]*
              D[u[x, t]*v[x, t], x]\  + \[IndentingNewLine]\[Epsilon]*v[x, t]*
              D[v[x, t], x]\  \[Equal] 
          0\[IndentingNewLine]}, \[IndentingNewLine]{u[x, t], \ 
        v[x, t]}, \[IndentingNewLine]{x, 
        t}, \[IndentingNewLine]{\[Alpha], \ \[Beta], \ \[Gamma]\ , \ \
\[Delta], \[Epsilon]}, \ \[IndentingNewLine]Form \[Rule] Sech, \ 
      Verbose \[Rule] True]\)], "Input"],

Cell[BoxData[
    \( (*\ 6.4\ Fisher\ and\ FitzHugh - 
        Nagumo\ \(\(Equations\)\(.\)\)\ *) \)], "Input"],

Cell[BoxData[
    \( (*\ Fisher\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(\(\(\ \)\(PDESpecialSolutions[{\[IndentingNewLine]D[u[x, t], 
              t]\  - \[IndentingNewLine]D[
              u[x, t], {x, 2}]\  - \[IndentingNewLine]u[x, 
                t]*\((1 - u[x, t])\)\  \[Equal] \ 
          0\[IndentingNewLine]}, \[IndentingNewLine]{u[x, t]}, {x, 
        t}, {}]\)\)\)], "Input"],

Cell[BoxData[
    \( (*\ FitzHugh - Nagumo\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{\[IndentingNewLine]beta*
              D[v[z], z]\  + \ \[IndentingNewLine]Sqrt[2]*
              D[v[z], \ {z, 2}]\  - \[IndentingNewLine]Sqrt[2]*
              v[z]*\((1 - Sqrt[2]*v[z])\)*\((alpha - 
                  Sqrt[2]*v[z])\)\  \[Equal] \ 
          0\[IndentingNewLine]}, \[IndentingNewLine]{v[
          z]}, \[IndentingNewLine]{z}, \[IndentingNewLine]{alpha, \ beta}, \ 
      NumericTest \[Rule] True]\)], "Input"],

Cell[BoxData[
    \( (*\ 6.5\ A\ Degenerate\ Hamiltonian\ System\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{\[IndentingNewLine]D[u[x, t], 
              t]\  - \[IndentingNewLine]D[u[x, t], 
              x]\  - \[IndentingNewLine]2*v[x, t]\  \[Equal] \ 
          0, \[IndentingNewLine]D[v[x, t], t]\  - \[IndentingNewLine]2*
              epsilon*u[x, t]*v[x, t]\  \[Equal] 
          0\[IndentingNewLine]}, \[IndentingNewLine]{u[x, t], v[x, t]}, {x, 
        t}, {epsilon}]\)], "Input"],

Cell[BoxData[
    \( (*\ 6.6\ The\ Combined\ KdV - mKdV\ equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t]\  + \[IndentingNewLine]6*alpha*u[x, t]*
            D[u[x, t], x]\  + \[IndentingNewLine]6*beta*u[x, t]^2*
            D[u[x, t], x]\  + \[IndentingNewLine]gamma*
            D[u[x, t], \ {x, 3}]\  \[Equal] 0, \[IndentingNewLine]u[x, 
        t], {x, t}, {alpha, \ beta, \ gamma}]\)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t]\  + \[IndentingNewLine]6*alpha*u[x, t]*
            D[u[x, t], x]\  + \[IndentingNewLine]6*beta*u[x, t]^2*
            D[u[x, t], x]\  + \[IndentingNewLine]gamma*
            D[u[x, t], \ {x, 3}]\  \[Equal] 0, \[IndentingNewLine]u[x, 
        t], {x, t}, {alpha, \ beta, \ gamma}, \ Form \[Rule] Sech]\)], "Input"],

Cell[BoxData[
    \( (*\ Duffing\ Equation\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[{D[u[x], \ {x, \ 2}]\  + \ u[x]\  + \ 
            alpha*u[x]^3\  \[Equal] \ 0}, \ u[x], \ {x}, \ {alpha}, \ 
      Form \[Rule] Sn, \ Verbose \[Rule] True, \ 
      SymbolicTest \[Rule] True]\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(PDESpecialSolutions[{D[
              u[x], \ {x, \ 2}]\  + \ u[x]\  + \ alpha*u[x]^3\  \[Equal] \ 
          0}, \ u[x], \ {x}, \ {alpha}, \ Form \[Rule] Cn, \ 
      Verbose \[Rule] True, \ SymbolicTest \[Rule] True]\)\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["China", "Subtitle"],

Cell[BoxData[
    \( (*\ 5.10\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t]\  + \ u[x, t]*D[u[x, t], x]\  + 
          p*D[u[x, t], \ {x, \ 3}]\  \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {p}, \ Form\  \[Rule] \ sn, \ 
      Verbose\  \[Rule] \ True, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ 5.16\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], {t, 2}]\  + 2*q*\ u[x, t]*D[u[x, t], {x, 2}]\  + \ 
          2*q*\((D[u[x, t], \ x])\)^2\  + \ p*D[u[x, t], \ {x, 2}]\  + \ 
          r*D[u[x, t], \ {x, \ 4}] \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {p, q, r}, \ Verbose\  \[Rule] \ True, \ 
      Form\  \[Rule] \ sn, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ 5.17\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t]\  + u[x, t]*D[u[x, t], x]\  + \ D[u[x, t], \ x]\  - \ 
          p*D[u[x, t], \ {x, 2}]\  - q*D[u[x, t], \ {x, \ 3}] \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {p, q}, \ Verbose\  \[Rule] \ True, \ 
      Form\  \[Rule] \ sn, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \(\(\( (*\ 
      5.17\ for\ tanh\ solutions\ \
*) \)\(\[IndentingNewLine]\)\(PDESpecialSolutions[
      D[u[x, t], t]\  + u[x, t]*D[u[x, t], x]\  + \ D[u[x, t], \ x]\  - \ 
          p*D[u[x, t], \ {x, 2}]\  - q*D[u[x, t], \ {x, \ 3}] \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {p, q}, \ Verbose\  \[Rule] \ True, \ 
      Form\  \[Rule] \ tanh, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)\)\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\( (*\ 
      5.24\ *) \)\(\[IndentingNewLine]\)\(PDESpecialSolutions[{D[u[x, t], 
              t]\  + u[x, t]*D[u[x, t], x]\  + \ D[v[x, t], \ x]\  + 
            s*D[u[x, t], \ {x, 2}] \[Equal] 
          0, \ \[IndentingNewLine]D[v[x, t], t]\  + u[x, t]*D[v[x, t], x] + 
            v[x, t]*D[u[x, t], x]\  + \ r*D[v[x, t], \ {x, 2}]\  + 
            p*D[u[x, t], \ {x, 3}] \[Equal] 0}, \ \[IndentingNewLine]\ {u[x, 
          t], v[x, t]}, \ {x, t}, \ {p, r, s}, \ Verbose\  \[Rule] \ True, \ 
      Form\  \[Rule] \ tanh, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)\)\)], "Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\( (*\ 
      5.29\ *) \)\(\[IndentingNewLine]\)\(PDESpecialSolutions[{D[u[x, t], 
              t]\  + u[x, t]*D[u[x, t], x]\  + \ D[v[x, t], \ x]\  + 
            q*D[D[u[x, t], \ {x, 2}], t] \[Equal] 
          0, \ \[IndentingNewLine]D[v[x, t], t]\  + u[x, t]*D[v[x, t], x] + 
            v[x, t]*D[u[x, t], x]\  + p*D[u[x, t], \ {x, 3}] \[Equal] 
          0}, \ \[IndentingNewLine]\ {u[x, t], v[x, t]}, \ {x, t}, \ {p, 
        q}, \ Verbose\  \[Rule] \ True, \ Form\  \[Rule] \ tanh, \ 
      NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)\)\)], "Input"],

Cell[BoxData[
    \(\(\( (*\ 
      5.29\ periodic\ *) \)\(\[IndentingNewLine]\)\(\(PDESpecialSolutions[{D[
              u[x, t], t]\  + u[x, t]*D[u[x, t], x]\  + \ D[v[x, t], \ x]\  + 
            q*D[D[u[x, t], \ {x, 2}], t] \[Equal] 
          0, \ \[IndentingNewLine]D[v[x, t], t]\  + u[x, t]*D[v[x, t], x] + 
            v[x, t]*D[u[x, t], x]\  + p*D[u[x, t], \ {x, 3}] \[Equal] 
          0}, \ \[IndentingNewLine]\ {u[x, t], v[x, t]}, \ {x, t}, \ {p, 
        q}, \ Verbose\  \[Rule] \ True, \ Form\  \[Rule] \ sn, \ 
      NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)\(\[IndentingNewLine]\)
    \)\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Abbott", "Subtitle"],

Cell[BoxData[
    \( (*\ Example\ 1\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t] + \ u[x, t]*D[u[x, t], x] - 
          D[u[x, t], \ {x, \ 5}]\  \[Equal] 0, \ u[x, t], \ {x, t}, \ {}, \ 
      Form\  \[Rule] \ sn, \ Verbose\  \[Rule] \ True, \ 
      NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 2\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t] + alpha*\ u[x, t]^2*D[u[x, t], x] - 
          D[u[x, t], \ {x, \ 5}]\  \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {alpha}, \ Form\  \[Rule] \ sn, \ 
      Verbose\  \[Rule] \ True, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 3\ \ Trouble\ with\ value\ of\ m[i] . \ 
          So\ give\ value\ of\ m[i]\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t] - sigma*u[x, t]*D[u[x, t], x] + u[x, t]^3*D[u[x, t], x] - 
          D[u[x, t], \ {x, \ 7}]\  \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {sigma}, \ Form\  \[Rule] \ sn, \ 
      Verbose\  \[Rule] \ True, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 4\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], {t, 2}] - gamma*\ D[u[x, t], {x, 2}] + alpha*u[x, t] - \ 
          beta*u[x, t]^2 \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {alpha, beta, \ gamma}, \ Form\  \[Rule] \ sn, \ 
      Verbose\  \[Rule] \ True, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 5\ *) \)], "Input"],

Cell[BoxData[
    \(PDESpecialSolutions[
      D[u[x, t], t] + u[x, t]*D[u[x, t], x] + D[u[x, t], {x, 2}] + 
          sigma*\[IndentingNewLine]D[u[x, t], {x, 3}] + 
          D[u[x, t], \ {x, \ 4}] + 
          epsilon*D[u[x, t]*D[u[x, t], x], x]\  \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {sigma, epsilon}, \ Form\  \[Rule] \ sn, \ 
      Verbose\  \[Rule] \ True, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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          D[u[x, t], {x, 5}] \[Equal] 0, \ u[x, t], \ {x, t}, \ {}, \ 
      Form\  \[Rule] \ sn, \ Verbose\  \[Rule] \ True, \ 
      NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 7\ *) \)], "Input"],

Cell[BoxData[
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          u[x, t]^2*D[u[x, t], x] + \[IndentingNewLine]D[u[x, t], {x, 3}] - 
          D[u[x, t], \ {x, \ 5}] \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {sigma}, \ Form\  \[Rule] \ sn, \ 
      Verbose\  \[Rule] \ True, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
    \( (*\ Example\ 8\ *) \)], "Input"],

Cell[BoxData[
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      D[u[x, t], t] + 
          D[6*u[x, t]^5 + 
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                    u[x, t]*D[u[x, t], x]^2)\) + D[u[x, t], {x, 4}], 
            x] \[Equal] 0, \ u[x, t], \ {x, t}, \ {alpha}, \ 
      Form\  \[Rule] \ sn, \ Verbose\  \[Rule] \ True, \ 
      NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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      D[u[x, t], t] + 
          D[20*alpha*u[x, t]^7 + 
              70*\((u[x, t]^4*\ D[u[x, t], {x, 2}] + 
                    2*u[x, t]^3*D[u[x, t], x]^2)\) + \[IndentingNewLine]14*
                alpha*\((u[x, t]^2*\ D[u[x, t], {x, 4}] + 
                    3*u[x, t]*D[u[x, t], {x, 2}]^2 + \[IndentingNewLine]4*
                      u[x, t]*\ D[u[x, t], x]*D[u[x, t], {x, 3}] + 
                    5*D[u[x, t], {x, 2}]*
                      D[u[x, t], x]^2)\) + \[IndentingNewLine]D[
                u[x, t], {x, 6}], x] \[Equal] 0, \ 
      u[x, t], \ {x, t}, \ {alpha}, \ Form\  \[Rule] \ sn, \ 
      Verbose\  \[Rule] \ True, \ NumericTest\  \[Rule] \ True, \ 
      SymbolicTest\  \[Rule] \ True]\)], "Input"]
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