2009/09/10 by Mathew A. Johnson, Johnson, Mathew A., Kevin Zumbrun +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.0909.1857
openalex publication_date 2009/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we investigate the spectral instability of periodic traveling\nwave solutions of the generalized Korteweg-de Vries equation to long wavelength\ntransverse perturbations in the generalized Kadomtsev-Petviashvili equation. By\nanalyzing high and low frequency limits of the appropriate periodic Evans\nfunction, we derive an orientation index which yields sufficient conditions for\nsuch an instability to occur. This index is geometric in nature and applies to\narbitrary periodic traveling waves with minor smoothness and convexity\nassumptions on the nonlinearity. Using the integrable structure of the ordinary\ndifferential equation governing the traveling wave profiles, we are then able\nto calculate the resulting orientation index for the elliptic function\nsolutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.\n