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Scattering theory for the Gross-Pitaevskii equation in three dimensions. (English) Zbl 1180.35481
This article is devoted to the Gross-Pitaevskii equation. This equation appears in the context of Bose-Einstein condensate and it is similar in form to the Ginzburg-Landau equation, to the Boussinesq equation and is sometimes referred to as a nonlinear Schrödinger equation. The authors are interested in the study of behavior of the solutions in both dispersive and non-dispersive regimes. They affirm that Gross-Pitaevskii equation has better dispersive properties than the wave equation (low frequency limit) and the Schrödinger equation (high frequency limit) in the context of the study. This is devoted to the behavior of small solutions in three dimensions. They prove that disturbances from the constant equilibrium with small, localize energy, disperse for large time, according to the linearized equation and that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it, a kind of asymptotic completeness theorem for the scattering related to this nonlinear equation and the bilinear interaction are considered in this work.
This article is the continuation of a previous work entitled “Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimension” by S. Gustafson, K. Nakanishi and Tai-Peng Tsai [Ann. Henri Poincaré 8, No. 7, 1303–1331 (2007; Zbl 1375.35485)], and it is also related to another one entitled “Scattering Theory for the Gross-Pitaevskii equation” by S. Gustafson, K. Nakanishi and Tai-Peng Tsai [Math. Res. Lett. 13, No. 2–3, 273–285 (2006; Zbl 1119.35084)].
In the article, there are not very clear references about the implications of the results achieved with respect to the physical background of the equation studied.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
81V80 Quantum optics
35B40 Asymptotic behavior of solutions to PDEs
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