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Rigorous resonant 1-d nonlinear geometric optics. (English) Zbl 0718.35094

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1990, No. 7, 12 p. (1990).
This brief note considers nonlinear generalizations of the usual asymptotic solutions of linear geometrical optics. Specifically, it is shown that the effect of nonlinearity is to introduce overtones (higher frequency components) into the analysis so that nonlinear problems do not usually have the familiar monochromatic solutions of linear theory. Single phase expansions involving special profiles are introduced and shown to describe the phenomena of the creation and interaction of these overtones. Their analysis rests on the theory of stratified solutions which are conormal with respect to the foliation by the level surfaces of the phase.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A60 Lasers, masers, optical bistability, nonlinear optics
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