Rarefaction pulses for the Nonlinear Schrodinger Equation in the transonic limit

TL;DR

This paper studies finite energy traveling waves of the nonlinear Schrödinger equation in 2D and 3D, showing they converge to Kadomtsev-Petviashvili ground states in the transonic limit and confirming a conjecture about traveling wave branches.

Contribution

It generalizes previous results to a broader class of nonlinearities and rigorously proves the existence of an upper branch of traveling waves in three dimensions.

Findings

Traveling waves converge to KP ground states in the transonic limit

Extension of results to a wider class of nonlinearities

Proof of existence of an upper branch of traveling waves in 3D

Abstract

We investigate the properties of finite energy travelling waves to the nonlinear Schrodinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev-Petviashvili equation. Our results generalize an earlier result of F. Bethuel, P. Gravejat and J-C. Saut for the two-dimensional Gross-Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of an upper branch of travelling waves in dimension three.