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March  2010, 9(2): 413-430. doi: 10.3934/cpaa.2010.9.413

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, Brazil
2. 

Instituto de Matemática Pura e Aplicada - IMPA, Estrada Dona Castorina, 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil

Received  April 2009 Revised  September 2009 Published  December 2009

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.
Keywords: Klein-Gordon-Schrödinger system., Periodic standing waves, orbital stability.
Mathematics Subject Classification: Primary: 35B10, 35B35; Secondary: 35Q9.
Citation: Fábio Natali, Ademir Pastor. Stability properties of periodic standing waves for the Klein-Gordon-Schrödinger system. Communications on Pure & Applied Analysis, 2010, 9 (2) : 413-430. doi: 10.3934/cpaa.2010.9.413
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