Stability and decay properties of solitary-wave solutions to the generalized BO--ZK equation

Abstract

Studied here is the generalized Benjamin-Ono--Zakharov-Kuznetsov equation $$ u_t+u^pu_x+\alpha\mathscr{H}u_{xx}+\varepsilon u_{xyy}=0, \quad (x,y)\in \mathbb R ^2,\;\;t\in \mathbb R ^+, $$ in two space dimensions. Here, $\mathscr{H}$ is the Hilbert transform and subscripts denote partial differentiation. We classify when this equation 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.