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Diffractive nonlinear geometric optics for short pulses. (English) Zbl 1036.35049

The paper deals with semilinear systems of hyperbolic partial differential equations on the time scale \(t= O(1/\varepsilon)\) being \(\varepsilon\ll 1\), the length of the pulse-like solutions. The amplitude is chosen so that nonlinear effects influence the leading term in the asymptotics.
For pulses of larger amplitude so that the nonlinear effects are pertinent for times \(t= O(1)\), accurate asymptotic solutions lead to transport equations similar to those valid in the case of wave trains. The opposite is true in the studied case. The profile equation for pulses for \(t= O(1/\varepsilon)\) is different from the associated equation for wave trains.

MSC:

35C20 Asymptotic expansions of solutions to PDEs
78A05 Geometric optics
35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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