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Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I

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Abstract

We consider the Zakharov equation in space dimension two

$$\left\{ {\begin{array}{*{20}c} {iu_t = - \Delta u + nu,} \\ {\frac{1}{{c_0^2 }}n_{tt} = \Delta n + \Delta \left| u \right|^2 } \\ \end{array} } \right.$$

. We prove the existence of blow-up solutions (stable “self-similar” blow-up solutions) for this problem and we study various properties of these solutions.

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Authors and Affiliations

  1. Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, Tour 55-65, 5ème étage, 4, place Jussieu, F-75252, Paris Cedex 05, France

    L. Glangetas

  2. Centre de Mathématiques, Universié de cergy-Pontoise, Avenue du Parc, 8, Le Campus, F-95033 Cergy, Pontoise Cedex, France

    F. Merle

Authors
  1. L. Glangetas
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  2. F. Merle
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Additional information

Communicated by T. Spencer

This work was partially done while the second author was visiting Rutgers University and Courant Institute

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Glangetas, L., Merle, F. Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I. Commun.Math. Phys. 160, 173–215 (1994). https://doi.org/10.1007/BF02099792

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  • DOI: https://doi.org/10.1007/BF02099792

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