Stability properties of periodic standing waves for theKlein-Gordon-Schrödinger system

Fábio Natali; Ademir Pastor

doi:10.3934/cpaa.2010.9.413

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2010, Volume 9, Issue 2: 413-430. Doi: 10.3934/cpaa.2010.9.413

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Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system

Fábio Natali^1, and
Ademir Pastor^2,

1.
Universidade Estadual de Maringá - UEM, Avenida Colombo, 5790, CEP 87020-900, Maringá
2.
Instituto de Matemática Pura e Aplicada - IMPA, Estrada Dona Castorina, 110, CEP 22460-320, Rio de Janeiro, RJ

Received: April 2009

Revised: September 2009

Published: March 2010

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Abstract

We study the existence and orbital stability/instability of periodic standing wave solutions for the Klein-Gordon-Schrödinger system with Yukawa and cubic interactions. We prove the existence of periodic waves depending on the Jacobian elliptic functions. For one hand, the approach used to obtain the stability results is the classical Grillakis, Shatah and Strauss theory in the periodic context. On the other hand, to show the instability results we employ a general criterium introduced by Grillakis, which get orbital instability from linear instability.

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Mathematics Subject Classification: Primary: 35B10, 35B35; Secondary: 35Q99.

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