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December 2009 Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation
Satoshi Masaki
Methods Appl. Anal. 16(4): 403-458 (December 2009).

Abstract

We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrödinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared. We also show by an analysis of the limit hydrodynamical equaiton that, however, this WKB-type approximation breaks down before reaching the focal point: Nonlinear effects lead to the formation of singularity of the leading term of the phase function.

Citation

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Satoshi Masaki . "Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation." Methods Appl. Anal. 16 (4) 403 - 458, December 2009.

Information

Published: December 2009
First available in Project Euclid: 12 October 2010

zbMATH: 1214.35065
MathSciNet: MR2734494

Subjects:
Primary: 35Q55
Secondary: 35Q31

Rights: Copyright © 2009 International Press of Boston

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Vol.16 • No. 4 • December 2009
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