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

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

2008-10-24

arxiv.org/abs/0810.4417v2

International Mathematics Research Notices 2009 (2009) 2700-2748

Mathematics

Analysis of PDEs

Final version accepted for publication in International Mathematics Research Notices with a few minor corrections and added re

Scientific paper

10.1093/imrn/rnp031

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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