Mathematics – Analysis of PDEs
Scientific paper
2009-09-10
Mathematics
Analysis of PDEs
26 pages. Sign error corrected in Lemma 3. Statement of main theorem corrected. Exposition updated and references added.
Scientific paper
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.
Johnson Mathew A.
Zumbrun Kevin
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