Coherent nonlinear waves and the Wiener algebra

Métivier, Guy; Joly, Jean-Luc; Rauch, Jeffrey

Métivier, Guy ; Joly, Jean-Luc ; Rauch, Jeffrey

Annales de l'Institut Fourier, Tome 44 (1994), p. 167-196 / Harvested from Numdam

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Abstract

On étudie les solutions oscillantes de systèmes semi linéaires du premier ordre $L u = f (t, x, u, \bar{u})$ , 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 $L u = f (t, x, u, \bar{u})$ , 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 = \sum_{τ, ω \in ℝ^{1 + d}} U_{τ, ω} (t, x) e^{i (τ T + ω . X)}, \sum ∥ U_{τ, ω} (t, x) ∥_{C ([0, t] : H^{s} (ℝ^{d}))} < \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 $u^{ε}$ by profile equations of standard from.

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

Publié le : 1994-01-01

MR 95c:35163 | Zbl 0791.35019 | 4 citations dans Numdam

DOI : https://doi.org/10.5802/aif.1393

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     author = {M\'etivier, Guy and Joly, Jean-Luc and Rauch, Jeffrey},
     title = {Coherent nonlinear waves and the Wiener algebra},
     journal = {Annales de l'Institut Fourier},
     volume = {44},
     year = {1994},
     pages = {167-196},
     doi = {10.5802/aif.1393},
     mrnumber = {95c:35163},
     zbl = {0791.35019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1994__44_1_167_0}
}

Métivier, Guy; Joly, Jean-Luc; Rauch, Jeffrey. Coherent nonlinear waves and the Wiener algebra. Annales de l'Institut Fourier, Tome 44 (1994) pp. 167-196. doi : 10.5802/aif.1393. http://gdmltest.u-ga.fr/item/AIF_1994__44_1_167_0/

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