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(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

uε=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,ω1+dUτ,ω(t,x)ei(τT+ω.X),Uτ,ω(t,x)C([0,t]:Hs(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.

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

uε=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.

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     title = {Coherent nonlinear waves and the {Wiener} algebra},
     journal = {Annales de l'Institut Fourier},
     pages = {167--196},
     publisher = {Institut Fourier},
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Métivier, Guy; Joly, Jean-Luc; Rauch, Jeffrey. 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/

[BW]M. Born and E. Wolf, Principles of optics, 4th ed., Pergamon Press, Oxford, 1970.

[CB]Y. Choquet-Bruhat, Ondes asymptotiques et approchées pour systèmes d'équations aux dérivées partielles non linéaires, J. Math. Pure Appl., 48 (1969), 117-158. | MR | Zbl

[C]R. Courant, Methods of mathematical physics, vol. II, Interscience Publishers, 1962. | Zbl

[D]J.-M. Delort, Oscillations semi-linéaires multiphasées compatibles en dimension 2 et 3 d'espace, J. Diff. Eq., 1991. | Zbl

[DM]R. Diperna and A. Majda, The validity of geometric optics for weak solutions of conservation laws, Comm. Math. Phys., 98 (1985), 313-347. | MR | Zbl

[G1]O. Gues, Développements asymptotiques de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal., 1993. | Zbl

[G2]O. Gues, Ondes multidimensionnelles epsilon stratifiées et oscillations, Duke Math. J., 1992. | MR | Zbl

[Hor]L. Hormander, The analysis of linear partial differential operators, vol. 1, Springer-Verlag, 1991.

[Hou]T. Hou, Homogenization for semilinear hyperbolic systems with oscillatory data, Comm. Pure Appl. Math., 41 (1988), 471-495. | MR | Zbl

[HK]J. Hunter and J. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math., 36 (1983), 547-569. | MR | Zbl

[HMR]J. Hunter, A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves II : several space variables, Stud. Appl. Math., 75 (1986), 187-226. | MR | Zbl

[JMR1] J.-L. Joly, G. Métivier and J. Rauch, Resonant one dimensional non linear geometric optics, J. Funct. Anal., 114 (1993), 106-231. | MR | Zbl

[JMR2]J.-L. Joly, G. Métivier and J. Rauch, Rigorous resonant 1−d nonlinear geometric optics, in Journées Équations aux Dérivées Partielles, St Jean de Monts, juin 1990, Publ. de l'École Polytechnique, Palaiseau. | EuDML | Numdam | MR | Zbl

[JMR3]J.-L. Joly, G. Métivier and J. Rauch, Formal and rigorous nonlinear high frequency hyperbolic waves, in Nonlinear Hyperbolic Equations and Field Theory, eds. M.K.V. Murthy and S. Spagnolo, Pitmann Research Notes in Mathematics #253, 1992, 121-144. | MR | Zbl

[JMR4]J.-L. Joly, G. Métivier and J. Rauch, Remarques sur l'optique géométrique non linéaire multidimensionnelle, Séminaire Équations aux Dérivées Partielles, École Polytechnique, exposé n° 1, 1990-1991. | EuDML | Numdam | MR | Zbl

[JMR5]J.-L. Joly, G. Métivier and J. Rauch, Coherent and focussing multidimensional nonlinear geometric optics, Annales de l'École Normale Supérieure, to appear. | EuDML | Numdam | MR | Zbl

[JMR6]J.-L. Joly, G. Métivier and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J., 70 (1993), 373-404. | MR | Zbl

[JMR7]J.-L. Joly, G. Métivier and J. Rauch, Nonlinear geometric optics with an oscillating plane, preprint.

[J]J.-L. Joly, Sur la propagation des oscillations semi-linéaires en dimension 1 d'espace, C.R. Acad. Sc. Paris, t. 296 (1983). | MR | Zbl

[JR1]J.-L. Joly and J. Rauch, Ondes oscillantes semi-linéaires en 1−d, in Journées Équations aux Dérivées Partielles, St Jean de Monts, juin 1986, Publ. de l'École Polytechnique, Palaiseau. | EuDML | Numdam | MR | Zbl

[JR2]J.-L. Joly and J. Rauch, Ondes oscillantes semi-linéaires à hautes fréquences, in Recent Developments in Hyperbolic Equations, (L. Cattabriga, F. Colombini, M. Murthy, S. Spagnolo, eds.), Pitman Research Notes in Math., 183 (1988), 103-115. | MR | Zbl

[JR3]J.-L. Joly and J. Rauch, High frequency semilinear oscillations, in Wave Motion : Theory, Modelling and Computation (A.-J. Chorin and A.-J. Majda, eds.), Springer-Verlag (1987), 202-217. | MR | Zbl

[JR4]J.-L. Joly and J. Rauch, Nonlinear resonance can create dense oscillations, in Microlocal Analysis and Nonlinear Waves, M. Beals, R. Melrose and J. Rauch eds.), Springer-Verlag (1991), 113-123. | MR | Zbl

[JR5]J.-L. Joly and J. Rauch, Justification of multidimensional single phase semilinear geometric optics, Trans. Amer. Math. Soc., 330 (1992), 599-625. | MR | Zbl

[KAl]L.-A. Kalyakin, Long wave asymptotics, integrable equations as asymptotic limit of nonlinear systems, Russian Math. Surveys, vol. 44, n° 1 (1989), 3-42. | MR | Zbl

[Kat]Y. Katznelson, An introduction to harmonic analysis 2d ed., Dover Publ., 1976. | MR | Zbl

[Kl]S. Klainerman, The null condition and global existence to nonlinear wave equations, Springer Lectures in Applied Mathematics, 23 (1986), 293-326. | MR | Zbl

[Kr]H.-O. Kreiss, Über sachgemässe cauchyprobleme, Math. Scand., 13 (1963), 109-128. | EuDML | MR | Zbl

[La]P.-D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646. | MR | Zbl

[Lu]D. Ludwig, Conical refraction in crystal optics and hydromagnetics, Comm. Pure Appl. Math., XIV (1961), 113-124. | MR | Zbl

[MR]A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves I : a single space variable, Stud. Appl. Math., 71 (1984), 149-179. | MR | Zbl

[MRS]A. Majda, R. Rosales and M. Schonbek, A canonical system of integrodifferential equations in nonlinear acoustics, Stud. Appl. Math., 79 (1988), 205-262. | MR | Zbl

[MPT]D. Maclaughlin, G. Papanicolau and L. Tartar, Weak limits of semilinear hyperbolic systems with oscillating data, Lecture Notes in Physics 230, Springer-Verlag (1985), 277-289. | MR | Zbl

[MU]R. Melrose and G. Uhlmann, Microlocal structure of involutive conical refraction, Duke Math. J., 46 (1979), 118-133. | MR | Zbl

[S]S. Schochet, Fast singular limits of hyperbolic partial differential equations, J. Diff. Eq., to appear. | MR | Zbl

[Tar]L. Tartar, Solutions oscillantes des équations de Carleman, Séminaire Goulaouic-Meyer-Schwartz, 1983. | Numdam | Zbl

[Tay]M. Taylor, Pseudodifferential operators, Princeton Mathematics Series #34, Princeton University Press, Princeton N.J., 1981. | MR | Zbl

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