SetDirectory["d:\\demoddespecialsolutions"]

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

Get["DDESpecialSolutionsV3.m"]

??  DDESpecialSolutions

(* In difficult cases,  
      there may be freedom in the choice of the degree of the polynomial solution .  Thus,  
      the software investigates all the possible solutions between MinDegreeOfThePolynomial and MaxDegreeOfThePolynomial.  *)

DDESpecialSolutions[{D[
            u[n, t],  {t, 
              2}]  ==  (1 + D[u[n, t],  t])*(u[n - 1, t] - 
                2*u[n, t] + u[n + 1, t])}, {u},  {n}, {t},  {},  
      Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t], t] - v[n, t] == 0,  
        D[v[n, t], 
              t] - (1 + v[n, t])*(u[n - 1, t] - 2*u[n, t] + 
                  u[n + 1, t]) == 0},  {u,  v},  {n}, {t},  {}, 
      Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t] , 
            t]  ==  (1 + alpha*u[n, t])*(v[n, t] - 
                v[n - 1, t]),  
        D[v[n, t],  t]  == 
          v[n, t]*(u[n + 1, t] - u[n, t] + alpha*v[n + 1, t] - 
                alpha*v[n - 1, t])},  {u, v}, {n}, {t}, {alpha}, 
      SymbolicTest -> True,  Verbose  ->  True]

DDESpecialSolutions[{Derivative[0, 1][u][n, t]==(1 + alpha u[n, t]) (-v[-1 + n, t] + 
                    v[n, t]),Derivative[0, 1][v][n, t]==v[n, 
                  t] (-u[n, t] + u[1 + n, t] - 
                    alpha v[-1 + n, t] + alpha v[1 + n, t])},{u, v},{n},{t},{alpha}, DegreeOfThePolynomial -> {m[1] -> 1, 
            m[2] -> 1}, Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t],  t]  ==  
          u[n, t]*(v[n, t] - v[n - 1, t]),  
        D[v[n, t],  t]  ==  
          v[n, t]*(u[n + 1, t] - u[n, t])},  {u,  
        v},  {n}, {t},  {},  InputForm -> True,  
      Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t],  t]  ==  
          u[n, t]*(v[n, t] - v[n - 1, t]),  
        D[v[n, t],  t]  ==  
          v[n, t]*(u[n + 1, t] - u[n, t])},  {u,  
        v},  {n}, {t},  {},  SolveAlgebraicSystem -> False,  
      Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t], 
            t] == (u[n + 1, t] - v[n, t])*
              v[n, t] - (u[n - 1, t] - v[n - 1, t])*v[n - 1, t], 
        D[v[n, t], t] == v[n, t]*(u[n + 1, t] - u[n, t])}, {u,  
        v}, {n}, {t}, {},  Verbose  ->  True]

(* 
      On some integrable systems related to the Toda equations *) 
 (* Ref :  Suris,  
      J . Phys . A . 30 (1997) 2235 *)

DDESpecialSolutions[{D[u[n, t], t] == 
          u[n, t]*(v[n, t] - v[n - 1, t]), 
        D[v[n, t], t] == v[n, t]*(u[n + 1, t] - u[n, t])}, {u, 
        v}, {n}, {t}, {},  Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t], t] - 
            alpha*(u[n + 1, t] - 2*u[n, t] + u[n - 1, t]) - 
            u[n, t]*v[n, t]*(u[n + 1, t] + u[n - 1, t]) == 0, 
        D[v[n, t], t] + alpha*(v[n + 1, t] - 2*v[n, t] + v[n - 1, t]) + 
            u[n, t]*v[n, t]*(v[n + 1, t] + v[n - 1, t]) == 
          0}, {u,  v},  {n}, {t},  {alpha},  
      Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t],  t]  ==  
          aa*(u[n + 1, t] - 2*u[n, t] + u[n - 1, t]) + 
            u[n, t]*v[n, t]*(u[n + 1, t] + u[n - 1, t]), 
        D[v[n, t], 
            t]  == -aa*(v[n + 1, t] - 2*v[n, t] + v[n - 1, t]) - 
            u[n, t]*v[n, t]*(v[n + 1, t] + v[n - 1, t])}, {u,  
        v}, {n}, {t}, {aa},  Verbose  ->  True]

DDESpecialSolutions[{D[u[n, t], t]  == 
          aa*(bb*u[n + 1, t] - 2*bb*u[n, t] + bb*u[n - 1, t]) + 
            u[n, t]*v[n, t]*(bb*u[n + 1, t] + bb*u[n - 1, t]), 
        D[v[n, t], 
            t]  == -aa*(bb*v[n + 1, t] - 2*bb*v[n, t] + 
                  bb*v[n - 1, t]) - 
            u[n, t]*v[n, t]*(bb*v[n + 1, t] + bb*v[n - 1, t])}, {u, 
        v}, {n}, {t}, {aa, bb},  Verbose  ->  True]

DDESpecialSolutions[
      D[u[n, t], 
          t] == (aa^2 - u[n, t]^2)*(u[n + 1, t] - u[n - 1, t]), 
      u, {n}, {t}, {aa},  Verbose  ->  True]

DDESpecialSolutions[{D[q[n, t], 
            t]  ==  (a + b*q[n, t] - 
                c*q[n, t]^2)*(r[n + 1, t]  -  r[n - 1, t]), 
        D[r[n, t],  
            t]  ==  (a + b*r[n, t]  -  
                c*r[n, t]^2)*(q[n + 1, t] - q[n - 1, t])}, {q, 
        r}, {n}, {t}, {a, b, c},  Verbose  ->  True]

(* 
      Self - dual nonlinear network equation *) 
 (* Time - 
        discretization of soliton equations by Ryogo Hirota and Masataka Iwao in SIDE III. *)

DDESpecialSolutions[
      D[v[n, t], 
          t]  ==  (1 - v[n, t]^2)*(v[n - 1, t]  -  
              v[n + 1, t]), v, n, t, {},  
      Verbose  ->  True]

DDESpecialSolutions[
      D[w[n, t], 
          t]  ==  (1 + b*w[n, t] + 
              c*w[n, t]^2)*(w[n - 1, t]  -  w[n + 1, t]), w, n, 
      t, {b, c},  Verbose  ->  True]

DDESpecialSolutions[
      D[s[n, x, t],  x, 
          t]  ==  (s[n - 1, x, t]  -  2*s[n, x, t] + 
              s[n + 1, x, t])*(D[s[n, x, t], t]  +  1), s, 
      n,  {x,  t}, {},  Verbose  ->  True]

DDESpecialSolutions[{D[u[n, x, t],  t]  ==  v[n, x, t],  
        D[v[n, x, t],  
            x]  ==  (u[n - 1, x, t]  -  2*u[n, x, t] + 
                u[n + 1, x, t])*(v[n, x, t]  +  1)}, {u, v}, 
      n,  {x,  t}, {},  Verbose  ->  True]