WKB analysis for the nonlinear Schrodinger equation and instability results

Details WKB analysis for the nonlinear Schrodinger equation and instability results WKB analysis for the nonlinear Schrodinger equation and instability results

2007-02-12

arxiv.org/abs/math/0702318v1

Mathematics

Analysis of PDEs

14 pages. Proceeding of conferences given at Sapporo, November 2006

Scientific paper

For the semi-classical limit of the cubic, defocusing nonlinear Schrodinger equation with an external potential, we explain the notion of criticality before a caustic is formed. In the sub-critical and critical cases, we justify the WKB approximation. In the super-critical case, the WKB analysis provides a new phenomenon for the (classical) cubic, defocusing nonlinear Schrodinger equation, which can be compared to the loss of regularity established for the nonlinear wave equation by G. Lebeau. We also show some instabilities at the semi-classical level.

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