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On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation. I. (English) Zbl 1183.35240
The one-dimensional Gross-Pitaevskii equation can be written in the form \[ i \partial_t \Psi + \partial^2_x \Psi = \Psi (|\Psi|^2 -1) , \] which is a version of the defocusing cubic nonlinear Schrödinger equation. It appears in various areas of physics such as fluid mechanics, nonlinear optics, and the Bose-Einstein condensation. It has been known for several years that the Korteweg-de Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation. In this paper the authors provide a rigorous proof of this fact and compute a precise estimate for the error term. Their proof uses the integrability of both equations. They also obtain a relation between the invariants of the two equations.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35A35 Theoretical approximation in context of PDEs
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