Why numerical methods diverge from analytical soliton solution of the nonlinear Schrodinger equation after some time $t$?

Question

After doing some simulations on soliton solutions of the non-linear Schrodinger equation of the form $$iu_t+u_{xx}+|u|^2u=0$$ whose soliton solution is $$u(x,t)=2 \operatorname{sech} \left( \sqrt{2} (x-2t) \right) \,e^{i(x-2t)},$$ I realized that numerical methods give an accurate solution only for some short time; after some time $t$ numerical solutions start becoming different from the analytical solution above, but it seems like the localized area remains unchanged - it's the surrounding area that seems to change. Why is that happening?

Answer 1

The non-linear term mixes the frequency components. In the numerical solution this happens slightly differently than in the exact solution, thus high-frequency components do not sum exactly and some high-frequency oscillation becomes visible.

In $$ u_t=i(u_xx+|u|^2u) $$ the time derivative is orthogonal to $u$. Thus $$ \frac12\partial_t\int |u(x,t)|^2dx=\int Re(\bar u u_t)dx =...+\int Re(-i|\bar u_x|^2+i|u|^2) =0 $$ or nearly so if the boundary values are small. This relation is better preserved by the solvers, so that the area remains (visibly) constant fora longer time.