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WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity. (English) Zbl 1167.35328
Summary: We consider the semi-classical limit for the Gross-Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. For a cubic-quintic nonlinearity, we show that working with analytic data may be necessary and sufficient to obtain a similar result.

MSC:
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
37L50 Noncompact semigroups; dispersive equations; perturbations of infinite-dimensional dissipative dynamical systems
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
82D50 Statistical mechanical studies of superfluids
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