On the Whitham system for the (2+1)-dimensional nonlinear Schr\"odinger equation

TL;DR

This paper derives the Whitham modulation equations for the 2D nonlinear Schrödinger equation with small dispersion, analyzes their structure, and investigates the stability of traveling waves, confirming instability through numerical methods.

Contribution

The paper presents the first derivation of the complete 2D NLS Whitham system in both physical and Riemann variables, including stability analysis of traveling waves.

Findings

Traveling waves are unstable in both elliptic and hyperbolic cases.

The derived modulation equations consist of six evolutionary equations and two constraints.

Numerical calculations support the theoretical stability results.

Abstract

We derive the Whitham modulation equations for the nonlinear Schr\"odinger equation in the plane (2d NLS) with small dispersion. The modulation equations are derived in terms of both physical and Riemann variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic case, the traveling waves are found to be unstable. This result is consistent with all previous investigations of such stability by other methods. These results are supported by direct numerical calculations.