Some nonlinear evolution equations in water waves
The paper is devoted to the study of two- and three-dimensional nonlinear water waves generated by a steady or oscillating pressure distribution acting at the free surface of a running stream on water of finite depth \(h\). The governing equations have the form: \[ \begin{aligned} & \nabla ^2\Phi =0\quad(-h(x,y)\leq z\leq 0,\quad x,y\in \mathbb{R}),\\ &\eta_t+U\eta_x + \Phi_x\eta_x+\Phi_y\eta_y-\Phi_z=0\text{ at }z=\eta,\\ &\Phi_t+U\Phi_x+g\eta +(1/2)(\nabla \Phi)^2-T(w_{1x}+w_{2y})=-p/\rho \text{ at }z=\eta ,\\ &\Phi_xh_x+\Phi_yh_y+\Phi_z=0\text{ on }z=-h(x,y),\end{aligned} \] where \(\Phi =\Phi (x,y,z,t)\) is the velocity potential, \(\eta =\eta (x,y,t)\) is the free surface elevation, the time \( t>0\), \(p\) is the pressure, \(g\) is the acceleration due to the gravity and \(w_i\) (\(i=1,2\)) the curvatures involved in the surface tension term. The stream moves in the \(x\)-direction with constant speed \(U\). The solutions of the corresponding linearized problems become singular depending on the relative values of the water depth, frequency, and the speed of the applied pressure leading to a resonant phenomenon. In the case if the applied oscillatory pressure distribution moves at a resonant speed, the finite amplitude response is governed by the forced nonlinear Schrödinger (fNLS) equation. Under certain circumstances, the generated wave disturbance may not reach a steady state; in particular, for deep water, a steady state is never attained. For the case of a two-dimensional wave generated by a localized steady pressure moving at a resonant speed, the generated waves are actually of bounded amplitude and are governed by a forced Korteweg-de Vries (fKdV) equation subject to appropriate asymptotic initial conditions. A computational study of the fKdV equation reveals that a set of solitons is generated in front of the pressure distribution. On the other hand, for the case of the three-dimensional waves induced by a localized steady pressure traveling at a resonant speed, the nonlinear response is governed by a forced Kadomtsev-Petviashvili (fKP) equation. In order to extend the range of applicability of the Boussinesq-type (BT) equations in the theory of water waves, the modified Boussinesq equations are derived in terms of the velocity potential on an arbitrary elevation and the free surface displacement.
scientific article; zbMATH DE number 1556928
scientific article