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

We study oscillatory solutions of semilinear first order symmetric hyperbolic system $Lu=f\left(t,x,u,\overline{u}\right)$, 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

 ${u}^{\epsilon }=U\left(t,x,t/\epsilon ,x/\epsilon \right)+o\left(1\right)$

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

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

 $U=\sum _{\tau ,\omega \in {ℝ}^{1+d}}{U}_{\tau ,\omega }\left(t,x\right){e}^{i\left(\tau T+\omega .X\right)},\sum \parallel {U}_{\tau ,\omega }\left(t,x\right){\parallel }_{C\left(\left[0,t\right]:{H}^{s}\left({ℝ}^{d}\right)\right)}<\infty .$

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 ${u}^{\epsilon }$ by profile equations of standard from.

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

On étudie les solutions oscillantes de systèmes semi linéaires du premier ordre $Lu=f\left(t,x,u,\overline{u}\right)$, 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

 ${u}^{\epsilon }=U\left(t,x,t/\epsilon ,x/\epsilon \right)+o\left(1\right).$

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

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.

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title = {Coherent nonlinear waves and the {Wiener} algebra},
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pages = {167--196},
publisher = {Institut Fourier},
volume = {44},
number = {1},
year = {1994},
doi = {10.5802/aif.1393},
mrnumber = {95c:35163},
zbl = {0791.35019},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.1393/}
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Joly, J.-L.; Métivier, G.; Rauch, J. Coherent nonlinear waves and the Wiener algebra. Annales de l'Institut Fourier, Volume 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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