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Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev–Petviashvili Equation
Johnson, Mathew A. ; Zumbrun, Kevin
Johnson, Mathew A.
Zumbrun, Kevin
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Abstract
In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev–Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles, we are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteweg–de Vries and modified Korteweg–de Vries equations.
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This is the published version, also available here: http://dx.doi.org/10.1137/090770758.
Date
2010-10-21
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Society for Industrial and Applied Mathematics
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Keywords
transverse instability, generalized Korteweg-de Vries equation, periodic waves, generalized Kadomtsev-Petviashvili equation
Citation
Johnson, Mathew A. & Zumbrun, Kevin. "Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev–Petviashvili Equation." SIAM J. Math. Anal., 42(6), 2681–2702. (22 pages). http://dx.doi.org/10.1137/090770758.