Conservation of phase space properties using exponential integrators on the cubic Schrödinger equation

Authors:
Håvard Berland;Alvaro L. Islas;Constance M. Schober
Affiliations:
Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway;Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA;Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA
Venue:
Journal of Computational Physics
Year:
2007

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Abstract

The cubic nonlinear Schrodinger (NLS) equation with periodic boundary conditions is solvable using Inverse Spectral Theory. The ''nonlinear'' spectrum of the associated Lax pair reveals topological properties of the NLS phase space that are difficult to assess by other means. In this paper we use the invariance of the nonlinear spectrum to examine the long time behavior of exponential and multisymplectic integrators as compared with the most commonly used split step approach. The initial condition used is a perturbation of the unstable plane wave solution, which is difficult to numerically resolve. Our findings indicate that the exponential integrators from the viewpoint of efficiency and speed have an edge over split step, while a lower order multisymplectic is not as accurate and too slow to compete.