Existence and stability of solitons for fully discrete approximations of the nonlinear Schr\"odinger equation

TL;DR

This paper investigates the existence and stability of numerical solitons in fully discrete approximations of the nonlinear Schrödinger equation, demonstrating long-time stability under certain conditions.

Contribution

It establishes conditions for the existence of numerical solitons close to continuous ones and proves long-time stability for symmetric initial data.

Findings

Existence of numerical solitons close to continuous solitons.

Long-time stability of numerical solutions near solitons.

Stability results depend on a CFL condition.

Abstract

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.