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

TL;DR

This paper rigorously proves that the Korteweg-de Vries equation accurately approximates long-wave, small-amplitude solutions of the 1D Gross-Pitaevskii equation, providing precise error estimates.

Contribution

It offers a rigorous proof of the long-standing approximation and relates invariants of both equations, enhancing theoretical understanding.

Findings

Rigorous proof of the KdV approximation for the Gross-Pitaevskii equation

Precise error estimates for the approximation

Relation between invariants of the two equations

Abstract

The fact that the Korteweg-de-Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature. In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.