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.
Categories: math.AP