Stability and Decay properties of Solitary wave solutions for the generalized BO-ZK equation

TL;DR

This paper investigates the existence, regularity, decay, and stability of solitary wave solutions for a two-dimensional generalized Benjamin-Ono--Zakharov-Kuznetsov equation, classifying solutions based on parameter signs and nonlinearity strength.

Contribution

It provides a comprehensive classification of solitary wave solutions, analyzes their properties, and studies their stability in the context of the generalized BO-ZK equation.

Findings

Solitary wave solutions exist under specific parameter conditions.

Regularity and decay properties are characterized.

Stability of solutions is established.

Abstract

Studied here is the generalized Benjamin-Ono--Zakharov-Kuznetsov equation $u_t+u^pu_x+\alpha\mathscr{H}u_{xx}+\varepsilon u_{xyy}=0, \quad (x,y)\in\rr^2\!,\;\;t\in \rr^+\!$ in two space dimensions. Here, $\mathscr{H}$ is the Hilbert transform and subscripts denote partial differentiation. We classify when equation (1) possesses solitary-wave solutions in terms of the signs of the constants $\alpha$ and $\varepsilon$ appearing in the dispersive terms and the strength of the nonlinearity. Regularity and decay properties of these solitary wave are determined and their stability is studied.