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

@article{AIF_1994__44_1_167_0,
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {1},
     year = {1994},
     pages = {167-196},
     doi = {10.5802/aif.1393},
     zbl = {0791.35019},
     mrnumber = {95c:35163},
     language = {en},
     url = {http://www.numdam.org/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, Volume 44 (1994) no. 1, pp. 167-196. doi : 10.5802/aif.1393. http://www.numdam.org/item/AIF_1994__44_1_167_0/

[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 41 #624 | Zbl 0177.36404

[C]R. Courant, Methods of mathematical physics, vol. II, Interscience Publishers, 1962. | Zbl 0099.29504

[D]J.-M. Delort, Oscillations semi-linéaires multiphasées compatibles en dimension 2 et 3 d'espace, J. Diff. Eq., 1991. | Zbl 0736.35001

[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 87e:35057 | Zbl 0582.35081

[G1]O. Gues, Développements asymptotiques de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal., 1993. | Zbl 0780.35017

[G2]O. Gues, Ondes multidimensionnelles epsilon stratifiées et oscillations, Duke Math. J., 1992. | MR 94a:35011 | Zbl 0837.35086

[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 89k:35136 | Zbl 0632.35039

[HK]J. Hunter and J. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math., 36 (1983), 547-569. | MR 716196 | MR 85h:35143 | Zbl 0547.35070

[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 867874 | MR 89h:35199 | Zbl 0657.35084

[JMR1] J.-L. Joly, G. Métivier and J. Rauch, Resonant one dimensional non linear geometric optics, J. Funct. Anal., 114 (1993), 106-231. | MR 1220985 | MR 94i:35118 | Zbl 0851.35023

[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. | Numdam | MR 1069957 | MR 92e:35104 | Zbl 0718.35094

[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 1175206 | MR 93j:35112 | Zbl 0824.35077

[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. | Numdam | MR 1131574 | Zbl 0749.35055

[JMR5]J.-L. Joly, G. Métivier and J. Rauch, Coherent and focussing multidimensional nonlinear geometric optics, Annales de l'École Normale Supérieure, to appear. | Numdam | MR 1305424 | Zbl 0836.35087

[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 1219817 | MR 94c:35048 | Zbl 0815.35066

[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 705687 | MR 84j:35112 | Zbl 0555.35081

[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. | Numdam | MR 874553 | Zbl 0614.35002

[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 984363 | MR 90d:35173 | Zbl 0724.35070

[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 920836 | MR 89c:35096 | Zbl 0703.35103

[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 1120286 | MR 92k:35180 | Zbl 0794.35098

[JR5]J.-L. Joly and J. Rauch, Justification of multidimensional single phase semilinear geometric optics, Trans. Amer. Math. Soc., 330 (1992), 599-625. | MR 1073774 | MR 92f:35040 | Zbl 0771.35010

[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 997682 | Zbl 0683.35082

[Kat]Y. Katznelson, An introduction to harmonic analysis 2d ed., Dover Publ., 1976. | MR 422992 | MR 54 #10976 | Zbl 0352.43001

[Kl]S. Klainerman, The null condition and global existence to nonlinear wave equations, Springer Lectures in Applied Mathematics, 23 (1986), 293-326. | MR 837683 | MR 87h:35217 | Zbl 0599.35105

[Kr]H.-O. Kreiss, Über sachgemässe cauchyprobleme, Math. Scand., 13 (1963), 109-128. | MR 168921 | MR 29 #6177 | Zbl 0145.13303

[La]P.-D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J., 24 (1957), 627-646. | MR 97628 | MR 20 #4096 | Zbl 0083.31801

[Lu]D. Ludwig, Conical refraction in crystal optics and hydromagnetics, Comm. Pure Appl. Math., XIV (1961), 113-124. | MR 127703 | MR 23 #B748 | Zbl 0112.21202

[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 760229 | MR 86e:35089 | Zbl 0572.76066

[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 975485 | MR 90g:76077 | Zbl 0669.76103

[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 815948 | Zbl 0588.76137

[MU]R. Melrose and G. Uhlmann, Microlocal structure of involutive conical refraction, Duke Math. J., 46 (1979), 118-133. | MR 544247 | MR 81b:58044 | Zbl 0422.58026

[S]S. Schochet, Fast singular limits of hyperbolic partial differential equations, J. Diff. Eq., to appear. | MR 1303036 | Zbl 0838.35071

[Tar]L. Tartar, Solutions oscillantes des équations de Carleman, Séminaire Goulaouic-Meyer-Schwartz, 1983. | Numdam | Zbl 0481.35010

[Tay]M. Taylor, Pseudodifferential operators, Princeton Mathematics Series #34, Princeton University Press, Princeton N.J., 1981. | MR 618463 | MR 82i:35172 | Zbl 0453.47026