Focusing of spherical nonlinear pulses in ℝ¹⁺³

Carles, R.; Rauch, J.

doi:10.1090/S0002-9939-01-06103-2

Focusing of spherical nonlinear pulses in ${\mathbb R}^{1+3}$
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Proc. Amer. Math. Soc. 130 (2002), 791-804 Request permission

Abstract:

This paper describes the behavior of spherical pulse solutions of semilinear wave equations in the limit of short wavelength. In three space dimensions we study the behavior of solutions which are described by nonlinear geometric optics away from the focal point. With a natural subcriticality hypothesis on the nonlinearity we prove that the possibly nonlinear effects at the focal point do not affect the usual description in terms of the Maslov index. That is, one has nonlinear geometric optics before and after the focal point with only the usual phase shift of $-1$. The reason is that the nonlinear effects occur on too small a set. We obtain a global asymptotic description which includes an approximation near the caustic, which is a solution of the free wave equation.

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