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Dana
10/08/08 07:36am

Dear all,

we have a problem with the DSolve operator. If we solve the following system of partial differential equations we get 10 solutions of which the sixth is of interest to us (as it is non-complex).

Solutionnoise =
FullSimplify[
DSolve[{-D[v[z, t], t] ==
D[v[z, t], z]^2/(2 s) + D[v[z, t], z] D[w[z, t], z]/s +
D[D[v[z, t], z], z]/2 g, -D[w[z, t], t] ==
D[w[z, t], z]^2/(2 s) + D[v[z, t], z] D[w[z, t], z]/s +
D[D[w[z, t], z], z]/2 g}, {v, w}, {z, t}]][[6]]

The result is v -> Function[{z, t}, C[4] + C[8] Tanh[z C[1] + t C[2] + C[3]]], w ->
Function[{z, t}, C[5] + C[9] Tanh[z C[1] + t C[2] + C[3]]].

If we now cross-check this by plugging it into the original dynamics we get something different from zero:

FullSimplify[
D[v[z, t], t] + D[v[z, t], z]^2/(2 s) +
D[v[z, t], z] D[w[z, t], z]/s + D[D[v[z, t], z], z]/2 g /.
Solutionnoise]
FullSimplify[
D[w[z, t], t] + D[w[z, t], z]^2/(2 s) +
D[v[z, t], z] D[w[z, t], z]/s + D[D[w[z, t], z], z]/2 g /.

Solutionnoise]. The output is:
(1/(2 s))C[8] Sech[
z C[1] + t C[2] + C[3]]^2 (2 s C[2] +
C[1]^2 ((C[8] + 2 C[9]) Sech[z C[1] + t C[2] + C[3]]^2 -
2 g s Tanh[z C[1] + t C[2] + C[3]])) and
(1/(2 s))C[9] Sech[
z C[1] + t C[2] + C[3]]^2 (2 s C[2] +
C[1]^2 ((2 C[8] + C[9]) Sech[z C[1] + t C[2] + C[3]]^2 -
2 g s Tanh[z C[1] + t C[2] + C[3]])).

How can this be? If it is really a solution, the dynamics should always be fulfilled.

Best;

Dana

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