On the modulation equations and stability of periodic generalized Korteweg-de Vries waves via Bloch decompositions

PHYSICA D-NONLINEAR PHENOMENA

卷:239

Article

Johnson, Mathew A.¹  Zumbrun, Kevin¹  Bronski, Jared C.² 

[1] Indiana Univ, Bloomington, IN 47405 USA

[2] Univ Illinois, Urbana, IL 61801 USA

关键词: Modulational instability;    Periodic waves;    Generalized Korteweg-de Vries equation;    Whitham equations;   

DOI  :  10.1016/j.physd.2010.07.012

来源: Elsevier

【 摘 要 】

In this paper, we consider the relation between Evans-function-based approaches to the stability of periodic travelling waves and other theories based on long-wavelength asymptotics together with Bloch wave expansions. In previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the linearized Whitham averaged system accurately describes the spectral stability to long-wavelength perturbations. To clarify the connection between Bloch-wave-based expansions and Evans-function-based approaches, we reproduce this result without reference to the Evans function by using direct Bloch expansion methods and spectral perturbation analysis. One of the novelties of this approach is that we are able to calculate the relevant Bloch waves explicitly for arbitrary finite-amplitude solutions. Furthermore, this approach has the advantage of being applicable in the more general multi-periodic setting where no conveniently computable Evans function has yet been devised. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free