Transverse spectral instabilities in Konopelchenko–Dubrovsky equation

Abstract

AbstractWe study the transverse spectral stability of the one‐dimensional small‐amplitude periodic traveling wave solutions of the (2+1)‐dimensional Konopelchenko–Dubrovsky (KD) equation. We show that these waves are transversely unstable with respect to two‐dimensional perturbations that are periodic in both directions with long wavelength in the transverse direction. We also show that these waves are transversely stable with respect to perturbations which are either mean‐zero periodic or square‐integrable in the direction of the propagation of the wave and periodic in the transverse direction with finite or short wavelength. We discuss the implications of these results for special cases of the KD equation—namely, KP‐II and mKP‐II equations.