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Semiclassical limit of the nonlinear Schrödinger-Poisson equation with subcritical initial data. (English) Zbl 1166.35374
Summary: We study the semi-classical limit of the nonlinear Schrodinger-Poisson (NLSP) equation for initial data of the WKB type. The semi-classical limit in this case is realized in terms of a density-velocity pair governed by the Euler-Poisson equations. Recently we have shown in [Indiana Univ. Math. J. 50, Spec. Iss., 109–157 (2001; Zbl 0989.35110)], that the isotropic Euler-Poisson equations admit a critical threshold phenomena, where initial data in the sub-critical regime give rise to globally smooth solutions. Consequently, we justify the semi-classical limit for sub-critical NLSP initial data and confirm the validity of the WKB method.

35Q55 NLS equations (nonlinear Schrödinger equations)
35B25 Singular perturbations in context of PDEs
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