On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation II

TL;DR

This paper analyzes the long-wave approximation of the one-dimensional Gross-Pitaevskii equation, showing solutions split into Korteweg-de Vries waves with error estimates, advancing understanding of wave behavior in quantum fluids.

Contribution

It extends previous analysis by providing improved error estimates for the Korteweg-de Vries approximation of the Gross-Pitaevskii equation at the long-wave limit.

Findings

Solutions split into two KdV waves with speeds ±√2

Error estimates are nearly optimal for traveling waves

Higher regularity improves approximation accuracy

Abstract

In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in a previous paper. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation split into two waves with opposite constant speeds $\pm \sqrt{2}$, each of which are solutions to a Korteweg-de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimate.