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

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.