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Orbital stability of traveling waves for the one-dimensional Gross-Pitaevskii equation. (English) Zbl 1232.35152
From the text: We prove the nonlinear orbital stability of the stationary traveling wave of the one-dimensional Gross-Pitaevskii equation (which models the dynamics of Bose-Einstein condensates, superfluids) by using Zakharov-Shabat’s inverse scattering method. The unusual conditions at infinity imposed by the finiteness of the Ginzburg-Landau energy give rise to the existence of many traveling waves solutions. Our main result is the nonlinear orbital stability for the solution \(U_0\), where \(U_c(x) = \sqrt{1-c^2/2} \tanh(\sqrt{1-c^2/2} x/\sqrt{2})+ i c/\sqrt{2}\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
35Q51 Soliton equations
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35C07 Traveling wave solutions
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