On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I
arXiv:0810.4417 · doi:10.1093/imrn/rnp031
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.
arXiv commentFinal version accepted for publication in International Mathematics Research Notices with a few minor corrections and added remarks