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December  2009, 25(4): 1163-1180. doi: 10.3934/dcds.2009.25.1163

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, United States
2. 

Department of Applied Mathematics, University of Washington, Campus Box 352420, Seattle, WA 98195, United States

Received  June 2008 Revised  May 2009 Published  September 2009

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.
Keywords: KdV equation, stability., cnoidal waves.
Mathematics Subject Classification: Primary: 35Q53, 37C75; Secondary: 35P1.
Citation: Nate Bottman, Bernard Deconinck. KdV cnoidal waves are spectrally stable. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1163-1180. doi: 10.3934/dcds.2009.25.1163
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