On the Whitham system for the radial nonlinear Schr\"odinger equation

TL;DR

This paper develops a Whitham modulation theory for the radial nonlinear Schrödinger equation, providing a unified approach to describe dispersive shock waves in two dimensions and comparing results with numerical simulations.

Contribution

It introduces a nonlinear WKB method to derive the Whitham system for the radial NLS, applicable to both integrable and nonintegrable cases, and compares the theory with numerical solutions.

Findings

Good quantitative agreement between Whitham theory and numerics.

Significant differences between 2D radial and 1D NLS DSW solutions.

Unified approach applicable to various PDEs.

Abstract

Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schr\"odinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose-Einstein condensates, water waves and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one-dimensional NLS equation exhibits significant qualitative and quantitative differences.