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On the KPI transonic limit of two-dimensional Gross-Pitaevskii travelling waves. (English) Zbl 1186.35199
The paper is dealing with the propagation of small-amplitude “sound waves” (density perturbations) in the Gross-Pitaevskii equation (GPE) with the repulsive nonlinear term,
\[ i\Psi_t = \Delta\Psi + (1 - |\Psi|^2)\Psi, \]
where \(\Delta\) is the multi-dimensional Laplacian. The sound wave propagates in the uniform finite-density state, which is described by solution \(\Psi = A \exp(-i(1-A^2)t\) with \(A\) = const (in fact, one may set \(A=1\)). In the limit case of the small-amplitude perturbation waves, one may perform the expansion of the GPE in powers of the amplitudes of the wave, up to the second order. This gives rise to the Kadomtsev-Petviashvili (KP-I) equation, which is integrable in the two-dimensional (2D) case (on the contrary to the nonintegrability of the underlying GPE). The paper reports a rigorous proof of the convergence of the small-amplitude perturbation-wave solutions of the two-dimensional GPE to ground-state solutions of the KP-I equation. In fact, this ground-state solution is the stable 2D weakly (algebraically, rather than exponentially) localized soliton of the “lump” type. The waves are transonic, i.e., they run at a velocity exceeding the sound velocity, which corresponds to the wave with an infinitely small amplitude.

35Q55 NLS equations (nonlinear Schrödinger equations)
76Q05 Hydro- and aero-acoustics
35C08 Soliton solutions
76H05 Transonic flows
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