Revista Matemática Iberoamericana


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Volume 20, Issue 3, 2004, pp. 815–864
DOI: 10.4171/RMI/408

Published online: 2004-12-31

Focusing of spherical nonlinear pulses in ${\mathbb R}^{1+3}$, II. Nonlinear caustic

Rémi Carles[1] and Jeffrey Rauch[2]

(1) Université Montpellier 2, France
(2) University of Michigan, Ann Arbor, USA

We study spherical pulse like families of solutions to semilinear wave equations in space time of dimension 1+3 as the pulses focus at a point and emerge outgoing. We emphasize the scales for which the incoming and outgoing waves behave linearly but the nonlinearity has a strong effect at the focus. The focus crossing is described by a scattering operator for the semilinear equation, which broadens the pulses. The relative errors in our approximate solutions are small in the $L^\infty$ norm.

Keywords: Geometric optics, short pulses, focusing, caustic, nonlinear scattering, high frequency asymptotics

Carles Rémi, Rauch Jeffrey: Focusing of spherical nonlinear pulses in ${\mathbb R}^{1+3}$, II. Nonlinear caustic. Rev. Mat. Iberoamericana 20 (2004), 815-864. doi: 10.4171/RMI/408