Rarefaction Waves of the Korteweg-de Vries Equation via Nonlinear Steepest Descent

TL;DR

This paper uses nonlinear steepest descent to analyze the long-time behavior of the Korteweg-de Vries equation with steplike initial data, revealing detailed asymptotics of rarefaction waves including previously unknown terms.

Contribution

It introduces a method to compute detailed asymptotics of rarefaction waves for the KdV equation, including higher-order terms, advancing understanding of wave behavior.

Findings

Derived the leading asymptotic behavior of rarefaction waves

Computed the next term in the asymptotic expansion

Enhanced the analytical understanding of long-time wave evolution

Abstract

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the next term in the asymptotic expansion of the rarefaction wave, which was not known before.