Remarks on solitary waves of the generalized two dimensional Benjamin-Ono equation
The author studies the two-dimensional generalized Benjamin-Ono equation \[ u_t+ \beta{\mathcal H}u_{xx}+\varepsilon v_y+ u^p u_x= 0,\quad u_y= V_x\tag{1} \] for \((x,y)\in \mathbb{R}^2\), \(t\geq 0\), where \(\varepsilon\) is a constant, \(\beta\in\mathbb{R}\), \({\mathcal H}\) is the Hilbert transform \[ {\mathcal H}u(x,y,t)=\text{p.v. }\pi^{-1} \int_{\mathbb{R}} {u(z,y,t)\over x-z}\,dt, \] p.v. being the Cauchy principal value. The author obtains the conditions of nonexistence of solitary wave solutions \(u\) of (1), proves the existence and regularity property of solitary wave solutions \(u\) of (1) which are continuous and belong to \(C^1_\infty(\mathbb{R}^2)\), studies variational properties of solutions and the nonlinear stability of the solitary waves.
scientific article; zbMATH DE number 5956388
scientific article