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Diffractive nonlinear geometric optics. (English) Zbl 0910.35076

Asymptotic problems which lead to Schrödinger type equations are considered in general form. The origin object is a constant coefficient symmetric hyperbolic system. Additional nonlinear terms in the equations are small on the considered small amplitude solution. A fast oscillating plane wave with a slow modulating small amplitude is taken as a leading order term of the asymptotic approximation on small parameter \(\varepsilon\). The main requirement is as follows: The transport equation for the wave amplitude is trivial, so that the leading term in linear geometric optics is a pure translation. It is well known this type asymptotics is valid over time interval order of unity as \(\varepsilon\to 0\). In the case under consideration it is possible to construct an asymptotics which is valid over a long time interval order of \(1/ \varepsilon\). To this end a slow deformation of the amplitude on the slow time \(t\varepsilon\) has to be found. Schrödinger type equations arise just in this way. Justification of the asymptotics is given in the paper. It should be noted that similar problems were previously analyzed by Russian mathematicians which the authors did not cite.
Reviewer: L.Kalyakin (Ufa)

MSC:

35L40 First-order hyperbolic systems
78A05 Geometric optics
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