Instability of double-periodic waves in the nonlinear Schrodinger equation

TL;DR

This paper develops a numerical method to compute the instability rates of double-periodic solutions in the cubic nonlinear Schrödinger equation, revealing their spectral instability and relation to the Lax spectrum.

Contribution

It introduces a simple numerical approach to analyze the spectral instability of double-periodic solutions in the NLS equation, extending understanding beyond standing waves.

Findings

Double-periodic solutions are spectrally unstable.

Instability relates to Lax spectrum bands outside the imaginary axis.

Numerical comparison shows differences in instability rates.

Abstract

It is shown how to compute the instability rates for the double-periodic solutions to the cubic NLS (nonlinear Schrodinger) equation by using the Lax linear equations. The wave function modulus of the double-periodic solutions is periodic both in space and time coordinates; such solutions generalize the standing waves which have the time-independent and space-periodic wave function modulus. Similar to other waves in the NLS equation, the double-periodic solutions are spectrally unstable and this instability is related to the bands of the Lax spectrum outside the imaginary axis. A simple numerical method is used to compute the unstable spectrum and to compare the instability rates of the double-periodic solutions with those of the standing periodic waves.