On the modulation equations and stability of periodic generalized Kortewegde Vries waves via Bloch decompositions

Mathew A. Johnson, Kevin Zumbrun, Jared C. Bronski

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)2057-2065
Number of pages9
JournalPhysica D: Nonlinear Phenomena
Volume239
Issue number23-24
DOIs
StatePublished - Nov 1 2010

Keywords

  • Generalized Kortewegde Vries equation
  • Modulational instability
  • Periodic waves
  • Whitham equations

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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