Whitham modulation theory for the defocusing nonlinear Schrodinger equation in two and three spatial dimensions

TL;DR

This paper derives multidimensional Whitham modulation equations for the defocusing nonlinear Schrödinger equation, enabling analysis of large amplitude wavetrains in applications like nonlinear photonics and matter waves.

Contribution

It introduces a novel derivation of Whitham equations in higher dimensions using a two-phase ansatz and period-averaging, extending previous one-dimensional results.

Findings

Derived multidimensional Whitham equations for NLS

Preserved invariance properties in the equations

Explicitly described limits and reductions

Abstract

The Whitham modulation equations for the defocusing nonlinear Schrodinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonics and matter waves.