A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation

Pages: 261 - 281, Volume 5, Issue 2, June 2012

doi:10.3934/krm.2012.5.261       Abstract        References        Full Text (477.2K)       Related Articles

Hua Chen - School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China (email)
Ling-Jun Wang - College of Science, Wuhan University of Science and Technology, Wuhan 430065, China (email)

Abstract: We study the spectral stability of the one-dimensional small amplitude periodic traveling wave solutions of the Zakharov-Kuznetsov equation with respect to two-dimensional perturbations, which are either periodic in the direction of propagation with the same period as the one-dimensional underlying traveling wave, or non-periodic (localized or bounded). Relying upon the perturbation theory for linear operators with periodic coefficients, we show that the small periodic traveling waves are transversely spectrally unstable, with respect to both types of perturbations.

Keywords:  ZK equation, spectral stability, small periodic traveling wave.
Mathematics Subject Classification:  Primary: 35Q53; Secondary: 37K45.

Received: March 2011;      Revised: December 2011;      Published: April 2012.

 References