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

Carles, Rémi; Rauch, Jeffrey

Rev. Mat. Iberoamericana, Tome 20 (2004) no. 1, p. 815-864 / Harvested from Project Euclid

Résumé

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.

Publié le : 2004-10-14
Classification: Geometric optics, short pulses, focusing, caustic, nonlinear scattering, high frequency asymptotics, 35B25, 35B33, 35B40, 35C20, 35L05, 35L60, 35L70, 35Q60, 78A45

@article{1098885436,
     author = {Carles, R\'emi and Rauch, Jeffrey},
     title = {Focusing of spherical nonlinear pulses in ${\mathbb R}^{1+3}$, II. Nonlinear caustic},
     journal = {Rev. Mat. Iberoamericana},
     volume = {20},
     number = {1},
     year = {2004},
     pages = { 815-864},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1098885436}
}

Carles, Rémi; Rauch, Jeffrey. Focusing of spherical nonlinear pulses in ${\mathbb R}^{1+3}$, II. Nonlinear caustic. Rev. Mat. Iberoamericana, Tome 20 (2004) no. 1, pp.  815-864. http://gdmltest.u-ga.fr/item/1098885436/