Transverse spectral stability of small periodic traveling waves for the KP equation

Details Transverse spectral stability of small periodic traveling waves for the KP equation Transverse spectral stability of small periodic traveling waves for the KP equation

2010-04-13

arxiv.org/abs/1004.2183v1

Mathematics

Analysis of PDEs

Scientific paper

The Kadomtsev-Petviashvili (KP) equation possesses a four-parameter family of one-dimensional periodic traveling waves. We study the spectral stability of the waves with small amplitude with respect to two-dimensional perturbations which are either periodic in the direction of propagation, with the same period as the one-dimensional traveling wave, or non-periodic (localized or bounded). We focus on the so-called KP-I equation (positive dispersion case), for which we show that these periodic waves are unstable with respect to both types of perturbations. Finally, we briefly discuss the KP-II equation, for which we show that these periodic waves are spectrally stable with respect to perturbations which are periodic in the direction of propagation, and have long wavelengths in the transverse direction.

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