Coherent nonlinear waves and the Wiener algebra

Joly, J.-L.; Métivier, G.; Rauch, J.

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 $Lu=f(t,x,u,\overline{u})$, 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

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

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 $Lu=f(t,x,u,\overline{u})$, 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

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

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

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

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.

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