Advances in Differential Equations

Nonlinear geometrical optics for oscillatory wave trains with a continuous oscillatory spectrum

David Lannes

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The frequency and the direction of propagation of an oscillatory wave train may be read on its oscillatory spectrum. Many works in geometrical optics allow the study of at most countable oscillatory spectra. In these works, the number of directions of propagation is therefore at most countable, while many physical effects would require a continuous infinity of directions of propagation. The goal of this paper is to make the nonlinear geometrical optics for wave trains with such a continuous oscillatory spectrum. This requires the introduction of new spaces, which are Wiener algebras associated to spaces of vector-valued measures with bounded total variation. We also make qualitative studies on the properties of wave trains with continuous oscillatory spectrum, and on the incidence of the nonlinearity on such oscillations. We finally suggest an application of the results of this paper to the study of both the spontaneous and the stimulated Raman scatterings.

Article information

Adv. Differential Equations, Volume 6, Number 6 (2001), 731-768.

First available in Project Euclid: 2 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q60: PDEs in connection with optics and electromagnetic theory
Secondary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35L45: Initial value problems for first-order hyperbolic systems


Lannes, David. Nonlinear geometrical optics for oscillatory wave trains with a continuous oscillatory spectrum. Adv. Differential Equations 6 (2001), no. 6, 731--768.

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