Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves. (English) Zbl 0815.35066

From the introduction: This note is devoted to the proof that, generically, formal oscillatory waves described by asymptotic expansions of infinite order using the WKB method, define rigorous approximate solutions close to exact oscillatory solutions of semilinear or quasilinear hyperbolic systems.


35L60 First-order nonlinear hyperbolic equations
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI


[1] V. Arnold, Chapitres supplémentaires de la théorie des équations différentielles ordinaires , “Mir”, Moscow, 1980. · Zbl 0956.34501
[2] Y. Choquet-Bruhat, Ondes asymptotiques et approchées pour des systèmes d’équations aux dérivées partielles non linéaires , J. Math. Pures Appl. (9) 48 (1969), 117-158. · Zbl 0177.36404
[3] J.-M. Delort, Oscillations semi-linéaires multiphasées compatibles en dimension \(2\) ou \(3\) d’espace , Comm. Partial Differential Equations 16 (1991), no. 4-5, 845-872. · Zbl 0736.35001
[4] O. Gues, Développements asymptotiques de solutions exactes de systèmes hyperboliques quasilinéaires , to appear in Asymptotic Anal.
[5] J. K. Hunter and J. B. Keller, Weakly nonlinear high frequency waves , Comm. Pure Appl. Math. 36 (1983), no. 5, 547-569. · Zbl 0547.35070
[6] J. K. Hunter, A. Majda, and R. Rosales, Resonantly interacting, weakly nonlinear hyperbolic waves. II. Several space variables , Stud. Appl. Math. 75 (1986), no. 3, 187-226. · Zbl 0657.35084
[7] J.-L. Joly, G. Métivier, and J. Rauch, Resonant one dimensional nonlinear geometric optics , to appear in J. Funct. Anal. · Zbl 0851.35023
[8] J.-L. Joly, G. Métivier, and J. Rauch, Nonlinear high frequency hyperbolic waves , Nonlinear hyperbolic equations and field theory (Lake Como, 1990) eds. M. K. Murthy and S. Spagnolo, Pitman Res. Notes Math. Ser., vol. 253, Longman Sci. Tech., Harlow, 1992, pp. 121-143. · Zbl 0824.35077
[9] J.-L. Joly, G. Métivier, and J. Rauch, Remarques sur l’optique géométrique non linéaire multidimensionnelle , Séminaire sur les Équations aux Dérivées Partielles, 1990-1991, École Polytech., Palaiseau, 1991, Exp. No. I, 17. · Zbl 0749.35055
[10] J.-L. Joly, G. Métivier, and J. Rauch, Coherent and focusing multidimensional nonlinear geometric optics , preprint, 1992. · Zbl 0836.35087
[11] J.-L. Joly, G. Métivier, and J. Rauch, Nonlinear waves with oscillating spectrum in a plane bundle , in preparation.
[12] J.-L. Joly, G. Métivier, and J. Rauch, Coherent nonlinear waves and the Wiener algebra , preprint, 1992.
[13] J.-L. Joly and J. Rauch, Justification of multidimensional single phase semilinear geometric optics , Trans. Amer. Math. Soc. 330 (1992), no. 2, 599-623. JSTOR: · Zbl 0771.35010
[14] J.-L. Joly and J. Rauch, Nonlinear resonance can create dense oscillations , Microlocal analysis and nonlinear waves (Minneapolis, MN, 1988-1989) eds. R. Melrose, M. Beals, and J. B. Rauch, IMA Vol. Math. Appl., vol. 30, Springer, New York, 1991, pp. 113-123. · Zbl 0794.35098
[15] L. A. Kalyakin, Long-wave asymptotics of the solution of a nonlinear system of equations with small dispersion , Differentsial’nye Uravneniya 23 (1987), no. 4, 696-705, 734. · Zbl 0672.35009
[16] L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium , Math. USSR-Sb. 60 (1988), 457-484. · Zbl 0699.35135
[17] L. A. Kalyakin, Long waves asymptotics. Integrable equations as asymptotic limits of nonlinear systems , Russian Math. Surveys 44 (1989), 3-42. · Zbl 0683.35082
[18] P. D. Lax, Asymptotic solutions of oscillatory initial value problems , Duke Math. J. 24 (1957), 627-646. · Zbl 0083.31801
[19] A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves. I. A single space variable , Stud. Appl. Math. 71 (1984), no. 2, 149-179. · Zbl 0572.76066
[20] V. P. Maslov and G. A. Omel’yanof, Interaction between small amplitude, short waves in a weakly dispersive plasma I , Ukranian Math. J. 39 (1987), 371-378. · Zbl 0701.76123
[21] S. Schochet, Fast singular limit of hyperbolic PDEs , preprint, 1992. · Zbl 0838.35071
[22] J.-C. Tougeron, Idéaux de fonctions différentiables , Ergeb. Math. Grenzgeb., vol. 71, Springer-Verlag, Berlin, 1972. · Zbl 0251.58001
[23] A. Yoshikawa, Solutions containing a large parameter of a quasilinear hyperbolic system of equations and their nonlinear geometric optics approximation , preprint, 1991.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.