Focusing at a point with caustic crossing for a class of nonlinear equations

2003/01/14 by Remi Carles, David Lannes, Carles, Remi +1
Mathematics · #35B33 #35B40 #35C20 #35L05 #35Q55 #78A05 #78A45 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B33 #msc:35B40 #msc:35C20 #msc:35L05 #msc:35Q55 #msc:78A05 #msc:78A45

paper · pdf · doi:10.48550/arxiv.math/0301144

25 pages

arxiv created 2003/01/14 · arxiv updated 2009/11/30

Abstract

We consider the asymptotic behavior of solutions to nonlinear partial differential equations in the limit of short wavelength. For initial data which cause focusing at one point, we highlight critical indexes as far as the influence of the nonlinearity in the neighborhood of the caustic is concerned. Our results generalize some previous ones proved by the first author in the case of nonlinear Schrodinger equations. We apply them to Hartree, Klein-Gordon and wave equations.

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