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

  • Yong Lu
  • Zhifei Zhang


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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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