Premium
Nonlinear oscillations beyond caustics
Author(s) -
Joly JeanLuc,
Métivier Guy,
Rauch Jeffrey
Publication year - 1996
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199605)49:5<443::aid-cpa1>3.0.co;2-b
Subject(s) - caustic (mathematics) , nonlinear system , mathematics , dissipative system , cotangent bundle , mathematical analysis , lift (data mining) , lipschitz continuity , trigonometric functions , classical mechanics , physics , geometry , quantum mechanics , computer science , data mining
This paper studies the focusing of high‐frequency solutions of semilinear hyperbolic equations. In previous papers, we studied two opposite phenomena. First, the focusing of nonlinear waves can force the solutions to blow up, even before reaching the caustics. Second, for strongly dissipative equations, nonlinear oscillations can be completely absorbed when they reach the caustic set. In this paper, we study the intermediate case of equations with globally Lipschitz nonlinearities. The nonlinear oscillations persist after crossing the caustic set. The solutions are described using oscillatory integrals, which are associated with Lagrangian manifolds in the cotangent bundle. The equations of nonlinear geometric optics lift to these manifolds. In contrast to the linear case, the transport equations for amplitudes living above the same points of spacetime are coupled. © 1996 John Wiley & Sons, Inc.