Coherent nonlinear waves and the Wiener algebra
Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 167-196.

On étudie les solutions oscillantes de systèmes semi linéaires du premier ordre Lu=f(t,x,u,u ¯), avec L hyperbolique symétrique et f fonction analytique réelle de ses arguments. La principale nouveauté est l’étude de problèmes multidimensionnels sous l’hypothèse naturelle de cohérence des phases, pour des profils presque périodiques non nécessairement quasi-périodiques. Par exemple, lorsque L est à coefficients constants et les phases sont linéaires, on construit des solutions qui possèdent une asymptotique haute fréquence de la forme

u ε = U ( t , x , t / ε , x / ε ) + o ( 1 ) .

L’analyse se fait avec des profils U(t,x,T,X) dans l’algèbre de Wiener des fonctions presque périodiques en (T,X).

Les profils sont uniquement déterminés à partir de leurs données initiales par le système habituel des équations de l’optique géométrique non linéaire.

L’analyse s’applique au cas des systèmes à caractéristiques de multiplicité variable, et permet de traiter l’exemple de la réfraction conique.

We study oscillatory solutions of semilinear first order symmetric hyperbolic system Lu=f(t,x,u,u ¯), with real analytic f.

The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in T,X with only the natural hypothesis of coherence.

In the special case where L has constant coefficients and the phases are linear, the solutions have asymptotic description

u ε = U ( t , x , t / ε , x / ε ) + o ( 1 )

where the profile U(t,x,T,X) is almost periodic in (T,X).

The main novelty in the analysis is the space of profiles which have the form

U = τ , ω 1 + d U τ , ω ( t , x ) e i ( τ T + ω . X ) , U τ , ω ( t , x ) C ( [ 0 , t ] : H s ( d ) ) < .

Thus, U is an element of the Wiener algebra as a function of the fast variables.

The profile U is uniquely determined from the initial data of u ε by profile equations of standard from.

An application to conical refraction where the characteristics have variable multiplicity is presented.

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     title = {Coherent nonlinear waves and the {Wiener} algebra},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Institut Fourier},
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Joly, J.-L.; Métivier, G.; Rauch, J. Coherent nonlinear waves and the Wiener algebra. Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 167-196. doi : 10.5802/aif.1393. http://archive.numdam.org/articles/10.5802/aif.1393/

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