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 norm.
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Rémi Carles, Jeffrey Rauch, Focusing of spherical nonlinear pulses in , II. Nonlinear caustic. Rev. Mat. Iberoam. 20 (2004), no. 3, pp. 815–864DOI 10.4171/RMI/408