KdV cnoidal waves are spectrally stable | Zendy

Nate Bottman | Zendy; Bernard Deconinck | Zendy

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KdV cnoidal waves are spectrally stable

Author(s) -

Nate Bottman,

Bernard Deconinck

Publication year - 2009

Publication title -

discrete and continuous dynamical systems

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.289

H-Index - 70

eISSN - 1553-5231

pISSN - 1078-0947

DOI - 10.3934/dcds.2009.25.1163

Subject(s) - korteweg–de vries equation , eigenfunction , stability (learning theory) , completeness (order theory) , mathematics , mathematical analysis , linear stability , physics , instability , computer science , nonlinear system , mechanics , eigenvalues and eigenvectors , quantum mechanics , machine learning

Going back to considerations of Benjamin (1974), there has been significant interest in the question of stability for the stationary periodic solu- tions of the Korteweg-deVries equation, the so-called cnoidal waves. In this pa- per, we exploit the squared-eigenfunction connection between the linear stabil- ity 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 eigenfunc- tions.

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