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Journal of Nonlinear Science

, Volume 3, Issue 1, pp 169–195 | Cite as

Nonelliptic Schrödinger equations

  • Jean-Michel Ghidaglia
  • Jean-Claude Saut
Article

Summary

Nonelliptic Schrödinger equations are defined as multidimensional nonlinear dispersive wave equations whose linear part in the space variables is not an elliptic equation. These equations arise in a natural fashion in several contexts in physics and fluid mechanics. The aim of this paper is twofold. First, a brief survey is made of the main nonelliptic Schrödinger equations known by the authors, with emphasis on water waves. Second, a theory is developed for the Cauchy problem for selected examples. The method is based on linear estimates which are strongly related to the dispersion relation of the problem.

Key words

nonlinear Schrödinger equations dispersive equations blow-up Cauchy problem 

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Copyright information

© Springer-Verlag New York Inc 1993

Authors and Affiliations

  • Jean-Michel Ghidaglia
    • 1
  • Jean-Claude Saut
    • 2
  1. 1.Centre de Mathématiques et leurs ApplicationsEcole Normale Supérieure de Cachan et CNRSCachan CedexFrance
  2. 2.Laboratoire d'Analyse NumériqueCNRS et Université Paris-SudOrsay CedexFrance

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