On the transverse instability of one dimensional capillary-gravity waves
The aim of this article is to prove the Lyapounov orbital instability of the line solitary water waves constructed by \textit{C. J. Amick} and \textit{K. Kirchgässner} [Arch. Ration. Mech. Analysis, 105, 1--49 (1989; Zbl 0666.76046)] governed by the system \[ \eta_{t}=\eta_{x}+G[\eta]\varphi, \] \[ \varphi_{t}=\varphi_{x}-\frac{1}{2}|\nabla\varphi|^{2}+\frac{1}{2}\frac{(G[\eta]\varphi+\nabla\varphi\cdot\nabla\eta)^{2}}{1+|\nabla\eta|^{2}}-\alpha\eta+\beta\nabla\cdot\frac{\nabla\eta}{\sqrt{1+|\nabla\eta|^{2}}} . \] Here \(\eta=\eta(t,x,y)\), \(\varphi=\varphi(t,x,y)\), \(t,x,y\in\mathbb{R}\), \(\nabla=(\partial_{x},\partial_{y})\) and \(\alpha=\frac{gh}{c^{2}}\), \(\beta=\frac{b}{hc^{2}},\) \(c\) represents the propagation speed, \(g\) the gravity constant, \(b\) takes into account the surface tension effects, \(h\) represents the deepness of the fluid domains and \(G[\eta]\) is a Dirichlet-Neumann map to be defined shortly with respect to transverse perturbations.
scientific article; zbMATH DE number 5715579