Methods and Applications of Analysis

Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation

Satoshi Masaki

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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.

Article information

Source
Methods Appl. Anal., Volume 16, Number 4 (2009), 403-458.

Dates
First available in Project Euclid: 12 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.maa/1286890988

Mathematical Reviews number (MathSciNet)
MR2734494

Zentralblatt MATH identifier
1214.35065

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35Q31: Euler equations [See also 76D05, 76D07, 76N10]

Keywords
Nonlinear Schrödinger equation semiclassical analysis WKB approximation caustics Euler equation

Citation

Masaki , Satoshi. Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation. Methods Appl. Anal. 16 (2009), no. 4, 403--458. https://projecteuclid.org/euclid.maa/1286890988


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