The formation of a soliton gas condensate for the focusing Nonlinear Schr\"odinger equation

TL;DR

This paper rigorously analyzes the formation of a soliton gas condensate in the focusing nonlinear Schrödinger equation as the number of solitons approaches infinity, revealing new configurations and confirming kinetic theory predictions.

Contribution

It introduces a rigorous analysis of multi-soliton solutions leading to soliton gas condensates, extending understanding beyond previous models with vanishing norming constants.

Findings

Soliton gas condensate forms as N approaches infinity with eigenvalues on two segments.

Solutions are described by rapidly oscillatory elliptic waves with constant velocity.

Rigorous justification of kinetic theory predictions in a deterministic setting.

Abstract

In this work, we carry out a rigorous analysis of a multi-soliton solution of the focusing nonlinear Schr\"{o}dinger equation as the number, $N$, of solitons grows to infinity. We discover configurations of $N$-soliton solutions which exhibit the formation (as $N \to \infty$) of a soliton gas condensate. Specifically, we show that when the eigenvalues of the Zakharov - Shabat operator for the NLS equation accumulate on two bounded horizontal segments in the complex plane with norming constants bounded away from $0$, then, asymptotically, the solution is described by a rapidly oscillatory elliptic-wave with constant velocity, on compact subsets of $(x,t)$. We then consider more complex solutions with an extra soliton component, and provide rigorous justification of the predictions of the kinetic theory of solitons in this deterministic setting. This is to be distinguished from previous analyses of soliton gasses where the norming constants were tending to zero with $N$, and the asymptotic description only included elliptic waves in the long-time asymptotics.