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

Hua Chen; Ling-Jun Wang

doi:10.3934/krm.2012.5.261

Article Contents

2012, Volume 5, Issue 2: 261-281. Doi: 10.3934/krm.2012.5.261

This issue Previous Article Boltzmann equation and hydrodynamics at the Burnett level Next Article Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension

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

Hua Chen^1, and
Ling-Jun Wang^2,

1.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072
2.
College of Science, Wuhan University of Science and Technology, Wuhan 430065

Received: February 28, 2011

Revised: November 30, 2011

Published: May 2012

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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.

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Mathematics Subject Classification: Primary: 35Q53; Secondary: 37K45.

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References

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