An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type

Jared C. Bronski, Mathew A. Johnson, Todd Kapitula

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg-de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L 2-critical KdV-4 equation that arises in the study of blow-up and the KdV-1/2 equation, which is an idealized model for plasmas.

Original languageEnglish (US)
Pages (from-to)1141-1173
Number of pages33
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume141
Issue number6
DOIs
StatePublished - Dec 2011

ASJC Scopus subject areas

  • Mathematics(all)

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