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{1286890988,
author = {Masaki , Satoshi},
title = {Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation},
journal = {Methods Appl. Anal.},
volume = {16},
number = {1},
year = {2009},
pages = { 403-458},
language = {en},
url = {http://dml.mathdoc.fr/item/1286890988}
}
Masaki , Satoshi. Cascade of Phase Shifts and Creation of Nonlinear Focal Points for Supercritical Semiclassical Hartree Equation. Methods Appl. Anal., Tome 16 (2009) no. 1, pp. 403-458. http://gdmltest.u-ga.fr/item/1286890988/