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Stability and decay properties of solitary-wave solutions to the generalized BO-ZK equation. (English) Zbl 1325.35162

Summary: Studied here is the generalized Benjamin-Ono-Zakharov-Kuznetsov equation \[ u_t+u^pu_x+\alpha\mathscr{H}u_{xx}+\epsilon u_{xyy}=0,\;\;\;(x,y)\in \mathbb{R}^2,\;t\in \mathbb{R}^+, \] 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 \(\epsilon\) 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.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35Q51 Soliton equations
35A15 Variational methods applied to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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