Local Existence and WKB Approximation of Solutions to Schrödinger-Poisson System in the Two-Dimensional Whole Space

We consider the Schrödinger-Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the...

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Bibliographic Details
Published inCommunications in partial differential equations Vol. 35; no. 12; pp. 2253 - 2278
Main Author Masaki, Satoshi
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis Group 04.11.2010
Taylor & Francis Ltd
Subjects
Mathematical analysis
Nonlinear equations
Nonlinear Schrödinger equation
Partial differential equations
Primary 35Q55
Schrodinger equation
Schrödinger equation
Secondary 81Q20
Semiclassical analysis
WKB approximation
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Summary:We consider the Schrödinger-Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schrödinger equation in the semiclassical limit.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605301003717142