KdV cnoidal waves are spectrally stable

Nate Bottman; Bernard Deconinck

doi:10.3934/dcds.2009.25.1163

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2009, Volume 25, Issue 4: 1163-1180. Doi: 10.3934/dcds.2009.25.1163

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KdV cnoidal waves are spectrally stable

Nate Bottman^1, and
Bernard Deconinck^2,

1.
Department of Applied Mathematics, University of Washington, Campus box 352420, Seattle, WA 98195
2.
Department of Applied Mathematics, University of Washington, Campus Box 352420, Seattle, WA 98195

Received: May 31, 2008

Revised: April 30, 2009

Published: October 2009

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Abstract

Going back to considerations of Benjamin (1974), there has been significant interest in the question of stability for the stationary periodic solutions of the Korteweg-deVries equation, the so-called cnoidal waves. In this paper, we exploit the squared-eigenfunction connection between the linear stability problem and the Lax pair for the Korteweg-deVries equation to completely determine the spectrum of the linear stability problem for perturbations that are bounded on the real line. We find that this spectrum is confined to the imaginary axis, leading to the conclusion of spectral stability. An additional argument allows us to conclude the completeness of the associated eigenfunctions.

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Mathematics Subject Classification: Primary: 35Q53, 37C75; Secondary: 35P10.

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