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arXiv · 0810.4417

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

Fabrice Bethuel·Philippe Gravejat·Jean-Claude Saut·Didier Smets
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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.

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Fabrice Bethuel, Philippe Gravejat, Jean-Claude Saut, Didier Smets. 2009-12-05. On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I. https://doi.org/10.1093/imrn%2Frnp031

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