. A P ] 1 2 Fe b 20 07 WKB ANALYSIS FOR THE NONLINEAR SCHRÖDINGER EQUATION AND INSTABILITY RESULTS
نویسنده
چکیده
For the semi-classical limit of the cubic, defocusing nonlinear Schrödinger 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 Schrödinger 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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