Spectral analysis of the periodic b-KP equation under transverse perturbations

The b-family-Kadomtsev–Petviashvili equation (b-KP) is a two dimensional generalization of the b-family equation. In this paper, we study the spectral stability of the one-dimensional small-amplitude periodic traveling waves with respect to two-dimensional perturbations which are either co-periodic in the direction of propagation, or nonperiodic (localized or bounded). We perform a detailed spectral analysis of the linearized problem associated to the above mentioned perturbations, and derive various stability and instability criteria which depends in a delicate way on the parameter value of b, the transverse dispersion parameter \(\sigma \) $\sigma$ , and the wave number k of the longitudinal waves.