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

Abstract

The cubic nonlinear Schrödinger (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.

Plane Waves Numerical Stability of Some Explicit Exponential Methods for Cubic Schrödinger Equation

“…which can be proved to be negative for every real value d. From here, the parabola is always negative for x > −2λ|a| 2 . This means that, for small enough h, the eigenvalues of the studied matrix have modulus < 1 for l 2 > −2λa 2 .…”

“…As it was stated in previous theorems, for every method considered in this paper, the stability or instability just depends on the values hλ|a| 2 and √ hl for each frequency and it is independent of the value k of the initial condition.…”

“…From this, it can be deduced that ϵ 0 (t) =ε 0 (0)e −λ|a| 2 it , which remains in modulus constant with time. On the other hand, for l ̸ = 0, as the eigenvalues of iG l are…”

“…It happens that the modulus of all the eigenvalues of the matrix in (3.6) are independent of k and just depend on λh|a| 2 and l 2 h. Furthermore, when…”

“…On the other hand, quite recently explicit exponential splitting and Runge-Kutta-based Lawson methods [2][3][4][5]9] have been thoroughly developed and recommended for cubic Schrödinger equation. The former conserve two invariants (norm and momentum) while the latter do not.…”