Effects of viscosity and surface tension on soliton dynamics in the generalized KdV equation for shallow water waves

Abstract

Various theories have been formulated for the study of weakly damped free-surface flows. These theories have been essentially focused on the forces relatively perpendicular to the fluid particle such as pressure forces, while neglecting forces relatively parallel to the fluid particle such as viscosity forces. In this work, with the help of linear approximation applying on the Navier-Stokes equations, we obtain a system of equations for potential flow which includes dissipative effect due to viscosity. The correction due to the viscosity is applied not only to the kinematics boundary condition on the surface, but also to the dynamics condition modeled by Bernoulli’s equation. We show that, in the context of wave motion in shallow water, an expansion of the Boussinesq system can be decomposed into a set of coupled equations. The first equation depends only on the surface elevation for the right-moving, while the other equation depends simultaneous on the surface elevation for the right- and left-moving waves. The wave equation corresponding to the pure right-moving has the form of a generalized inhomogeneous Korteweg de Vries (KdV) equation with higher-order nonlinear and dissipative terms. We then investigate the soliton solutions of this equation by using the Hirota’s bilinear method. The results show that, both group and phase velocities are a decreasing functions of the viscosity and surface tension parameters, δ and τ, respectively. The width of the soliton increases with the parameters δ and τ. The effects of viscosity on the soliton dynamics are more pronounced and are amplified by the surface tension effects.