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Rarefaction pulses for the nonlinear Schrödinger equation in the transonic limit. (English) Zbl 1292.35274
Authors’ abstract: We investigate the properties of finite energy travelling waves to the nonlinear Schrödinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev-Petviashvili equation. Our results generalize an earlier result of Béthuel et al. for the two-dimensional Gross-Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of vortexless travelling waves with high energy and momentum in dimension three.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35C07 Traveling wave solutions
76H05 Transonic flows
35Q53 KdV equations (Korteweg-de Vries equations)
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