Existence and stability of solitons for fully discrete approximations of the nonlinear Schrödinger equation

Details Existence and stability of solitons for fully discrete approximations of the nonlinear Schrödinger equation Existence and stability of solitons for fully discrete approximations of the nonlinear Schrödinger equation

2012-03-28

arxiv.org/abs/1203.6183v1

Mathematics

Numerical Analysis

2012-25

Scientific paper

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schr\"odinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schr\"odinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical soliton which is close in energy norm to the continuous soliton. Such result is valid under a CFL condition between the time and space stepsizes. Furthermore we prove that if the initial datum is symmetric and close to the continuous soliton, then the associated numerical solution remains close to the orbit of the continuous soliton for very long times.

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