Spectral analysis of periodic $b$-KP equation under transverse perturbation

TL;DR

This paper investigates the spectral stability of small-amplitude periodic traveling waves in the $b$-KP equation under transverse perturbations, revealing stability criteria dependent on parameters $b$, $\sigma$, and $k$.

Contribution

It provides a detailed spectral analysis and stability criteria for the $b$-KP equation's periodic waves under two-dimensional perturbations, a novel extension of previous one-dimensional studies.

Findings

Stability depends on $b$, $\sigma$, and $k$ parameters.

Derived explicit stability and instability conditions.

Identified parameter regimes for wave stability.

Abstract

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 $\sigma$, and the wave number $k$ of the longitudinal waves.