SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Plenary Lecture


Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation

L. Gauckler, E. Faou and C. Lubich

Abstract
The cubic nonlinear Schrödinger equation has solutions that are plane waves. In the talk we will discuss the stability of these solutions and the stability of their numerical approximation.
We first study the stability of plane waves in the exact solution. We show orbital stability of plane waves over long times. In the second part of the talk we study a very popular method for the numerical discretization of the nonlinear Schrödinger equation, the split-step Fourier method. This method combines a Fourier spectral method in space with a splitting integrator in time. We will pursue the question whether the stability of plane waves in the exact solution transfers to this numerical discretization.

Bibliography
[1] E. Faou, L. Gauckler and Ch. Lubich, Sobolev stability of plane wave solutions to the cubic nonlinear Schrödinger equation on a torus, Comm. Partial Differential Equations, 38 (2013), pp. 1123-1140.
[2] E. Faou, L. Gauckler and Ch. Lubich, Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation, preprint, arXiv:1306.0656.

Organized by         Universidad de Valladolid     IMUVA