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 language||English (US)|
|Number of pages||33|
|Journal||Proceedings of the Royal Society of Edinburgh Section A: Mathematics|
|State||Published - Dec 2011|
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