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

Published online by Cambridge University Press:  15 November 2011

Jared C. Bronski ,
Mathew A. Johnson  and
Todd Kapitula
Jared C. Bronski
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA (bronski@illinois.edu)
Mathew A. Johnson
Affiliation:
Department of Mathematics, Indiana University, 831 East 3rd Street, Bloomington, IN 47405, USA
Todd Kapitula
Affiliation:
Department of Mathematics and Statistics, Calvin College, 1740 Knollcrest Circle SE, Grand Rapids, MI 49546, USA
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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 L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 6 , December 2011 , pp. 1141 - 1173
Copyright
Copyright © Royal Society of Edinburgh 2011

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