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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Focusing of spherical nonlinear pulses in ${\mathbb R}^{1+3}$


Authors: R. Carles and J. Rauch
Journal: Proc. Amer. Math. Soc. 130 (2002), 791-804
MSC (2000): Primary 35B40, 35C20, 35L05, 35Q60
Published electronically: August 29, 2001
MathSciNet review: 1866035
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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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Additional Information

R. Carles
Affiliation: Département Mathématiques et Informatique, Antenne de Bretagne de l’ENS Cachan and IRMAR (Université Rennes 1), Campus de Ker Lann, 35 170 Bruz, France
Email: carles@bretagne.ens-cachan.fr

J. Rauch
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: rauch@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06103-2
PII: S 0002-9939(01)06103-2
Received by editor(s): September 15, 2000
Published electronically: August 29, 2001
Additional Notes: This research was begun while the first author was visiting the University of Michigan and he would like to thank that institution for its hospitality
The second author’s research was partially supported by U.S. National Science Foundation grant NSF-DMS-9803296
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society