Modulated solitons algorithm for nonlinear Schrödinger equations

We discuss new algorithms for the nonlinear cubic Schrödinger equation, based on the representation by modulated solitons in interactions. We consider the general idea of modulation and calculate explicitly the evolution of a single soliton for which the evolution system of the modulation parameters forms a completely integrable system. We then discuss the numerical approximation of nonlinear interactions of several modulated solitons through the Dirac-Frenkel variational principle. Numerical examples illustrate that the method yields very satisfying results in the case of weak interactions, and allows the computations of interacting solitons in the whole space without a full grid of discretization.

Book section

To appear in Recent progress on numerical analysis for nonlinear dispersive equations, published by World Scientific.