Welcome to the PDESpecialSolutions Package for Mathematica 4.1:
By Douglas Baldwin, Unal Goktas and Willy Hereman
© 2001

SetDirectory["c:\\myDirectory"]

<< PDESpecialSolutions.m

??  PDESpecialSolutions

The Boussinesq system is a simple system of two equations with one parameter alpha which yields a tanh solution.  The default option on form is Tanh.  This system also exhibits freedom in the leading order analysis;  however, the freedom doesn't lead to any additional solutions.

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},  InputForm -> False,  
      Verbose -> True,  NumericTest  ->  True,  
      SymbolicTest  ->  True]

PDESpecialSolutions[{Derivative[0, 1][u][x, t]+Derivative[1, 0][v][x, t]==0,Derivative[0, 1][v][x, t]+Derivative[1, 0][u][x, t]-3 u[x, t] Derivative[1, 0][u][x, t]-alpha Derivative[3, 0][u][x, t]==0},{u[x, t], v[x, t]},{x, t},{alpha},InputForm -> False,Verbose -> True]

Modified 3D KdV is an example of a single nonlinear PDE which yields two Sech solution.

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,  
      Test -> True,  SymbolicTest  ->  True]

Gao and Tian is a system of three equations which admit numerous mixed solutions of the mixed Sech-Tanh case.  This one takes a while.  In the mixed case, constant solutions are not removed.

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]

Duffing Equation (ODE) yields solutions of the Jacobi elliptic CN and SN functions.  Both are shown.  In the Jacobi elliptic function options, constant solutions are not removed.

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

PDESpecialSolutions[{D[u[x], {x, 2}] + u[x] + alpha*u[x]^3 == 
          0},  u[x], {x}, {alpha},  Form -> Sn,  
      InputForm -> False,  
NumericTest  ->  
        True,  SymbolicTest  ->  True]