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

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.

Final version accepted for publication in Communications in Partial Differential Equations with a few minor corrections and added remarks

References in corpus (2)

• Large time existence for 3D water-waves and asymptotics
• Orbital stability of the black soliton to the Gross-Pitaevskii equation

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