Archive for Rational Mechanics and Analysis

, Volume 222, Issue 1, pp 245–283 | Cite as

Partially Strong Transparency Conditions and a Singular Localization Method In Geometric Optics



This paper focuses on the stability analysis of WKB approximate solutions in geometric optics with the absence of strong transparency conditions under the terminology of Joly, Métivier and Rauch. We introduce a compatible condition and a singular localization method which allows us to prove the stability of WKB solutions over long time intervals. This compatible condition is weaker than the strong transparency condition. The singular localization method allows us to do delicate analysis near resonances. As an application, we show the long time approximation of Klein–Gordon equations by Schrödinger equations in the non-relativistic limit regime.


Cauchy Problem Geometric Optic Gordon Equation Singular Localization Transparency Condition 


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  1. 1.
    Bao W., Dong X.: Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime. Numer. Math. 120, 189–229 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cheverry C., Guès O., Métivier G.: Oscillations fortes sur un champ linéairement dégénéré. Ann. Sci. Ecole Normale Sup. 36, 691–745 (2003)MATHGoogle Scholar
  3. 3.
    Colin T., Lannes D.: Long-wave short-wave resonance for nonlinear geometric optics. Duke Math. J. 107, 351–419 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Colin T., Lannes D.: Justification of and long-wave correction to Davey–Stewartson systems from quadratic hyperbolic systems. Discrete Contin. Dyn. Syst. 11, 83–100 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Colin T., Ebrard G., Gallice G., Texier B.: Justification of the Zakharov model from Klein–Gordon-waves systems. Commun. Partial Differ. Equ. 29, 1365–1401 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Donnat, P., Joly, J.-L., Métivier, G., Rauch, J.: Diffractive nonlinear geometric optics, Séminaire Equations aux Dérivées Partielles. Ecole Polytechnique, Palaiseau, vol. XVII, p. 25, 1995–1996Google Scholar
  7. 7.
    Dumas E.: About nonlinear geometric optics. Bol. Soc. Esp. Mat. Apl. SeMA 35, 7–41 (2006)MathSciNetMATHGoogle Scholar
  8. 8.
    Ginibre J., Velo G.: The global Cauchy problem for the nonlinear Klein–Gordon equation. Math. Z. 189, 487–505 (1985)MathSciNetMATHGoogle Scholar
  9. 9.
    Ginibre J., Velo G.: The global Cauchy problem for the nonlinear Klein–Gordon equation II. Ann. Inst. H. Poincaré Anal. Non Linéaire 6, 15–35 (1989)MathSciNetMATHGoogle Scholar
  10. 10.
    Joly J.-L., Métivier G., Rauch J.: Coherent and focusing multidimensional nonlinear geometric optics. Ann. Sci. Ecole Norm. Sup. 28, 51–113 (1995)MathSciNetMATHGoogle Scholar
  11. 11.
    Joly J.-L., Métivier G., Rauch J.: Diffractive nonlinear geometric optics with rectification. Indiana Univ. Math. J. 47, 1167–1241 (1998)MathSciNetMATHGoogle Scholar
  12. 12.
    Joly J.-L., Métivier G., Rauch J.: Transparent nonlinear geometric optics and Maxwell–Bloch equations. J. Differ. Equ. 166, 175–250 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Klainerman S.: Global existence of small amplitude solutions to nonlinear Klein–Gordon equations in four space-time dimensions. Commun. Pure Appl. Math. 38, 631–641 (1985)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lannes D., Rauch J.: Validity of nonlinear geometric optics with times growing logarithmically. Proc. Am. Math. Soc. 129, 1087–1096 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lax P.D.: Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24, 627–646 (1957)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lu Y.: High-frequency limit of the Maxwell–Landau–Lifshitz system in the diffractive optics regime. Asymptotic Anal. 82, 109–137 (2013)MathSciNetMATHGoogle Scholar
  17. 17.
    Lu Y.: Higher-order resonances and instability of high-frequency WKB solutions. J. Differ. Equ. 260, 2296–2353 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lu, Y., Texier, B.: A stability criterion for high-frequency oscillations. Mém. Soc. Math. Fr. 142, vi+130 (2015)Google Scholar
  19. 19.
    Lu, Y., Zhang, Z.: Higher order asymptotic analysis of the nonlinear Klein–Gordon equation in the non-relativistic limit regime (2015) (preprint)Google Scholar
  20. 20.
    Majda, A.: Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables. Applied Math Sciences, vol. 53. Springer, New York, 1984Google Scholar
  21. 21.
    Masmoudi N., Nakanishi K.: From nonlinear Klein–Gordon equation to a system of coupled nonliner Schrödinger equations. Math. Ann. 324, 359–389 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Métivier, G.: Para-Differential Calculus and Applications to the Cauchy Problem for Nonlinear Systems. Centro di Ricerca Matematica Ennio De Giorgi, CRM Series, vol. 5. Edizioni della Normale, Pisa, pp. xii+140, 2008Google Scholar
  23. 23.
    Shatah J.: Normal forms and quadratic nonlinear Klein–Gordon equations. Commun. Pure Appl. Math. 38, 685–696 (1985)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Texier B.: The short wave limit for nonlinear, symmetric hyperbolic systems. Adv. Differ. Equ. 9, 1–52 (2004)MathSciNetMATHGoogle Scholar
  25. 25.
    Texier B.: Derivation of the Zakharov equations. Arch. Ration. Mech. Anal. 184, 121–183 (2007)MathSciNetCrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematical Institute, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic
  2. 2.School of Mathematical SciencePeking UniversityBeijingPeople’s Republic of China

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