Question: Travelling wave reduction of a PDE

Dear Maple Community,

I would like to ask you a question which will certainly be elementary for you. Imagine that I have a PDE (or, more generally, a system of PDEs) with (t, x) being independent variables, and the dependent variable defining the wave height or the fluid particle velocity u(t,x). The best example is the famous KdV equation:

u[t] + u*u[x] + u[x,x,x] = 0.

Now, I would like to automatically derive the ODE(s) that satisfy the travelling waves of this equation. Namely, we have to substitute the travelling wave ansatz u(t,x) = U(X) = U(x - c*t), where c is the travelling wave speed. In the case of the KdV equation, we obtain the following ODE:

-c*U' + U*U' + U''' = 0,

where prime ' denotes the derivative with respect to the new variable X.

My question is the following: What is the best way to automatically obtain this PDE -> ODE reduction in Maple?

Thanks a lot in advance!

Kind regards,


Please Wait...