Methods and Applications of Analysis
- Methods Appl. Anal.
- Volume 16, Number 4 (2009), 403-458.
Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation
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.
Methods Appl. Anal., Volume 16, Number 4 (2009), 403-458.
First available in Project Euclid: 12 October 2010
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35Q31: Euler equations [See also 76D05, 76D07, 76N10]
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