Focusing of Spherical Nonlinear Pulses In

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 eeects at the focal point do not aaect 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 eeects 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. 1. Introduction Many recent works have investigated the algorithms of nonlinear geometric optics (see 8] for a survey). These algorithms construct approximate solutions which are accurate when the wavelength denoted by " tends to zero. Most of these articles are valid for wave trains, satisfying the so-called slowly varying envelope approximation. It has been known for a long time that the slowly varying envelope approximation can be violated when studying ultrashort laser pulses (see e.g. 11]). In that case, wave trains are replaced with pulses with length O("). The mathematical study of such pulses is recent, and the construction of correctors and justiication of the approximation are diierent from the analogous problem for the propagation of wave trains (see 1], 2]). In the present article, we give a rst result analyzing what happens when short pulses pass through a focal point. We study semilinear wave equations in three space dimensions, with a subcritical nonlinearity. For wavetrains it is known that the caustic does not change the leading order wave train asymptotics away from the caustic (see 7], 10], 9], 3]). We analyse spherical pulses (Figure1), and show that nonlinear geometric optics, as constructed in 2], is valid away from the caustic. At the focus, the approximation is not good, since the exact solution is a regular function, whereas the proole of geometric optics is singular. However, this phenomenon occurs in so small a region that the approximation is valid before and after the caustic. As in 6] or in 4], the leading term of the approximation satisses a nonlinear transport equation away from the focal point. In a small neighborhood of the focal …