SetDirectory["e:\\data\\demopdespecialsolutions\\\
analyzesolvemarch2010"]

Get["PDESpecialSolutionsV3.m"]

??  PDESpecialSolutions

PDESpecialSolutions[
      D[u[x, t],  t]  +  alpha*u[x, t]*D[u[x, t], x]  +  
          D[u[x, t], {x, 3}]  ==  0,  
      u[x, t],  {x, t},  {alpha},  Verbose  ->  True,  
      Form  ->  Tanh,  SymbolicTest  ->  True,  
      NumericTest  ->  True]

PDESpecialSolutions[
      D[u[x, t],  t]  +  alpha*u[x, t]*D[u[x, t], x]  +  
          D[u[x, t], {x, 3}]  ==  0,  
      u[x, t],  {x, t},  {alpha},  Verbose  ->  True,  
      Form  ->  Tanh,  SymbolicTest  ->  False,  
      NumericTest  ->  False]

generalpde5thorder[x_, t_] :=  
      D[u[x, t], t] + alpha*u[x, t]^2*D[u[x, t], x] + 
        beta*D[u[x, t], x]*D[u[x, t], {x, 2}] + 
        gamma*u[x, t]*D[u[x, t], {x, 3}] + D[u[x, t], {x, 5}]

generalpde5thorder[x, t]

Laxpde5thorder[x_, 
        t_] :=   generalpde5thorder[x, t]  /. {alpha  ->  30,  
          beta ->  20,  gamma  ->  10}

Laxpde5thorder[x, t]

PDESpecialSolutions[Laxpde5thorder[x, t] ==  0,  
      u[x, t],  {x, t},  {},  Verbose  ->  True]

SKpde5thorder[x_, 
        t_] :=   generalpde5thorder[x, t]  /. {alpha  ->  5,  
          beta ->  5,  gamma  ->  5}

SKpde5thorder[x, t]

PDESpecialSolutions[SKpde5thorder[x, t] ==  0,  
      u[x, t],  {x, t},  {},  Verbose  ->  True,  
      SymbolicTest  ->  True,  
      NumericTest  ->  True]

KKpde5thorder[x_, 
        t_] :=   generalpde5thorder[x, t]  /. {alpha  ->  20,  
          beta ->  25,  gamma  ->  10}

KKpde5thorder[x, t]

PDESpecialSolutions[KKpde5thorder[x, t] ==  0,  
      u[x, t],  {x, t},  {},  Verbose  ->  True]

Itopde5thorder[x_, 
        t_] :=   generalpde5thorder[x, t]  /. {alpha  ->  2,  
          beta ->  6,  gamma  ->  3}

Itopde5thorder[x, t]

PDESpecialSolutions[Itopde5thorder[x, t] ==  0,  
      u[x, t],  {x, t},  {},  Verbose  ->  True,  
      NumericTest  ->  False,  
      SymbolicTest  ->  True]

(* 
      START OF A STUDY OF SEVENTH ORDER KDV TYPE EQUATIONS *)

generalpde7thorder[x_, t_] :=  
      D[u[x, t], t] + a*u[x, t]^3*D[u[x, t], x] + b*D[u[x, t], x]^3 + 
        c*u[x, t]*D[u[x, t], x]*D[u[x, t], {x, 2}] + 
        d*u[x, t]^2*D[u[x, t], {x, 3}] + 
        e*D[u[x, t], {x, 2}]*D[u[x, t], {x, 3}] + 
        f*D[u[x, t], {x, 1}]*D[u[x, t], {x, 4}] + 
        g*u[x, t]*D[u[x, t], {x, 5}] + D[u[x, t], {x, 7}]

Laxpde7thorder[x_, 
        t_] :=   generalpde7thorder[x, t]  /. {a  ->  140,  
          b ->  70,  c ->  280,  d ->  70,  
          e  ->  70,  f  ->  42,  
          g  ->  14}

Laxpde7thorder[x, t]

PDESpecialSolutions[Laxpde7thorder[x, t] ==  0,  
      u[x, t],  {x, t},  {},  Verbose  ->  True,  
      NumericTest  ->  True,  
      SymbolicTest  ->  True]

SKpde7thorder[x_, 
        t_] :=   generalpde7thorder[x, t]  /. {a  ->  252,  
          b ->  63,  c ->  378,  d ->  126,  
          e  ->  63,  f  ->  42,  
          g  ->  21}

SKpde7thorder[x, t]

PDESpecialSolutions[SKpde7thorder[x, t] ==  0,  
      u[x, t],  {x, t},  {},  Verbose  ->  True]

KKpde7thorder[x_, 
        t_] :=   generalpde7thorder[x, t]  /. {a  ->  2016,  
          b ->  630,  c ->  2268,  d ->  504,  
          e  ->  252,  f  ->  147,  
          g  ->  42}

KKpde7thorder[x, t]

PDESpecialSolutions[KKpde7thorder[x, t] ==  0,  
      u[x, t],  {x, t},  {},  Verbose  ->  True]

PDESpecialSolutions[
      D[u[x, t],  t]  +  alpha*u[x, t]*D[u[x, t], x]  +  
          D[u[x, t], {x, 3}]  ==  0,  
      u[x, t],  {x, t},  {alpha},  Form -> sech,  
      Verbose  ->  True]

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}]  ==  0},  {u[x, t]},  {x, 
        t},  {alpha},  Verbose -> True]

PDESpecialSolutions[{D[u[x, t], t] + D[v[x, t], x]  ==  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}]  ==  0}, {u[x, t], 
        v[x, t]},  {x, t},  {alpha},  Verbose  ->  True]

PDESpecialSolutions[{D[u[x, t], t] + D[v[x, t], x]  ==  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}]  ==  0},  {u[x, t], 
        v[x, t]},  {x, t},  {alpha}, 
      SolveAlgebraicSystem -> False]

PDESpecialSolutions[{D[u[x, t], t]  - 
            alpha * (6 * u[x, t] * D[u[x, t], x]   +   D[
                    u[x, t],  {x, 3}])  + 
            2 * beta * v[x, t]* D[v[x, t],  x]   == 0, 
        D[v[x, t], t]  +  3*u[x, t]*D[v[x, t], x]  + 
            D[v[x, t], {x, 3}]  ==  0}, {u[x, t],  v[x, t]}, {x, 
        t}, {alpha,  beta},  Form -> Sech,  
      Verbose  ->  True]

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}]  ==  0,  
      u[x, y, z, t],  {x, y, z, t},  {},  Form -> Sech,  
      Verbose  ->  True]

PDESpecialSolutions[{D[u[x, t],  t]  -  D[u[x, t],  x]  -  
            2*v[x, t]  ==  0,  
        D[v[x, t],  t]  +  2*u[x, t]*w[x, t]  ==  0,  
        D[w[x, t],  t]  +  2*u[x, t]*v[x, t]  ==  0},  {u[x, 
          t],  v[x, t],  w[x, t]},  {x, t},  {},  
      Form -> SechTanh]

PDESpecialSolutions[
      D[u[x, y, z, t], t]  +  alpha*u[x, y, z, t]*D[u[x, y, z, t], x]  + 
          D[u[x, y, z, t],  {x, 3}] + D[u[x, y, z, t], x,  y, y]  + 
          D[u[x, y, z, t],  x,  z,  z]  == 0, 
      u[x, y, z, t], {x, y, z, t}, {alpha},  
      Verbose  ->  True]

PDESpecialSolutions[
      D[u[x, y, z, t], t]  +  alpha*u[x, y, z, t]*D[u[x, y, z, t], x]  + 
          D[u[x, y, z, t],  {x, 3}] + D[u[x, y, z, t],  x,  y, y]  + 
          D[u[x, y, z, t],  x,  z,  z]  == 0, 
      u[x, y, z, t], {x, y, z, t}, 
{alpha},  
      Form -> Sech,  Verbose  ->  True]

PDESpecialSolutions[
      D[u[x, y, z, t],  t]  + 
          alpha * u[x, y, z, t] ^ 2 * D[u[x, y, z, t], x]  +  
          D[u[x, y, z, t],  {x, 3}] + D[u[x, y, z, t],  x,  y, y]  + 
          D[u[x, y, z, t],  x,  z,  z]  == 0, 
      u[x, y, z, t], {x, y, z, t}, {alpha},  
      Verbose  ->  True]

PDESpecialSolutions[
      D[u[x, y, z, t],  t]  + 
          alpha * u[x, y, z, t] ^ 2 * D[u[x, y, z, t], x]  +  
          D[u[x, y, z, t],  {x, 3}] + D[u[x, y, z, t],  x,  y, y]  + 
          D[u[x, y, z, t],  x,  z,  z]  == 0, 
      u[x, y, z, t], {x, y, z, t}, {alpha},  Form -> Sech,  
      Verbose  ->  True]

(* Example 9 :  
        Generalized Kuramoto - 
          Sivashinsky Equation           GOOD TEST CASE *)

PDESpecialSolutions[
      D[u[x, t], t]  +  u[x, t]* D[u[x, t], x]  +  D[u[x, t], {x, 2}] + 
          alpha* D[u[x, t], {x, 3}]  +  D[u[x, t], {x, 4}]  ==  0, 
      u[x, t], {x, t}, {alpha},  Verbose  ->  True]

PDESpecialSolutions[{D[u[x, t], t]  - 
            alpha * (6 * u[x, t] * D[u[x, t], x]   +   D[
                    u[x, t],  {x, 3}])  + 
            2 * beta * v[x, t]* D[v[x, t],  x]   == 0, 
        D[v[x, t], t]  +  3*u[x, t]*D[v[x, t], x]  + 
            D[v[x, t], {x, 3}]  ==  0}, {u[x, t],  v[x, t]}, {x, 
        t}, {alpha,  beta},  Form -> Cn,  
      Verbose  ->  True]

(* Example 11 :  
        Another Coupled KdV Systems (due to Guha and Roy) *)

PDESpecialSolutions[{D[u[x, t], t]  +   α * v[x, t]*
              D[v[x, t], x]  +  β*u[x, t]*
              D[u[x, t], x]  + γ * D[u[x, t], {x, 3}]  == 0, 
        D[v[x, t], t]  +  δ*D[u[x, t]*v[x, t], x]  + ϵ*
              v[x, t]*D[v[x, t], x]  == 0}, {u[x, t],  v[x, t]}, {x, 
        t}, {α,  β,  γ ,  δ, ϵ},  
      Form -> Sech,  Verbose  ->  True]

PDESpecialSolutions[{D[u[x, t], t]  - D[u[x, t], {x, 2}]  - 
            u[x, t]*(1 - u[x, t])  ==  0}, {u[x, t]}, {x, 
        t}, {},  Verbose  ->  True]

PDESpecialSolutions[{beta*D[v[z], z]  +  
            Sqrt[2]*D[v[z],  {z, 2}]  - 
            Sqrt[2]*v[
                z]*(1 - Sqrt[2]*v[z])*(alpha - 
                  Sqrt[2]*v[z])  ==  0}, {v[z]}, {z}, {alpha,  
        beta},  Verbose  ->  True]

PDESpecialSolutions[{D[u[x, t], t]  - D[u[x, t], x]  - 
            2*v[x, t]  ==  0, 
        D[v[x, t], t]  - 2*epsilon*u[x, t]*v[x, t]  == 0}, {u[x, t], 
        v[x, t]}, {x, t}, {epsilon},  Form -> SechTanh,  
      Verbose  ->  True]

PDESpecialSolutions[
      D[u[x, t], t]  + 6*alpha*u[x, t]*D[u[x, t], x]  + 
          6*beta*u[x, t]^2*D[u[x, t], x]  + 
          gamma*D[u[x, t],  {x, 3}]  == 0, 
      u[x, t], {x, t}, {alpha,  beta,  gamma},  
      Verbose  ->  True]

PDESpecialSolutions[
      D[u[x, t], t]  + 6*alpha*u[x, t]*D[u[x, t], x]  + 
          6*beta*u[x, t]^2*D[u[x, t], x]  + 
          gamma*D[u[x, t],  {x, 3}]  == 0, 
      u[x, t], {x, t}, {alpha,  beta,  gamma},  Form -> SechTanh,  
      Verbose  ->  True]

PDESpecialSolutions[{D[u[x],  {x,  2}]  +  u[x]  -  
            alpha*u[x]^3  ==  0},  u[x],  {x},  {alpha},  
      Form -> Cn,  Verbose  ->  True]

PDESpecialSolutions[{D[u[x],  {x,  2}]  +  u[x]  -  
            alpha*u[x]^3  ==  0},  u[x],  {x},  {alpha},  
      Form -> Tanh,  Verbose  ->  True]

generalizedfisherequation  =   u[x, t]*D[u[x, t], t]  - 
        u[x, t]*D[u[x, t], {x, 2}]  - 
        D[u[x, t], 
            x]^2 - u[x, t]^2*(1 - u[x, t]^3)

PDESpecialSolutions[{generalizedfisherequation  == 0},  
      u[x, t],  {x, t},  {},  Form -> Tanh,  
      Verbose  ->  True]

transformedgeneralizedfisherequation  =  
      generalizedfisherequation  /. 
{u[x, 
              t]   ->  
            v[x, t]^(2/3),  Derivative[m_, n_][u][x, 
              t]  ->  D[v[x, t]^(2/3), {x, m}, {t, n}]}

transformedfisherequation  = 
      Numerator[ 
        Factor[transformedgeneralizedfisherequation]]

PDESpecialSolutions[{transformedfisherequation  == 0},  
      v[x, t],  {x, t},  {},  Form -> Tanh,  
      Verbose  ->  True]