Abstract
The present study deals with the head-on collision process between capillary–gravity solitary waves in a finite channel. The present mathematical modeling is based on Nwogu’s Boussinesq model. This model is suitable for both shallow and deep water waves. We have considered the surface tension effects. To examine the asymptotic behavior, we employed the Poincaré–Lighthill–Kuo method. The resulting series solutions are given up to third-order approximation. The physical features are discussed for wave speed, head-on collision profile, maximum run-up, distortion profile, the velocity at the bottom, and phase shift profile, etc. A comparison is also given as a particular case in our study. According to the results, it is noticed that the free parameter and the surface tension tend to decline the solitary-wave profile significantly. However, the maximum run-up amplitude was affected in great measure due to the surface tension and the free parameter.
Highlights
1 Introduction The Korteweg–de Vries (KdV) equation [1] is used to represent the asymptotic behavior of weakly nonlinear two-dimensional unidirectional solitary waves in shallow water
Gardner et al [2] have shown the exact solution for the solitary waves with the help of the inverse scattering transform (IST) scheme
7 Conclusions This article deals with the head-on collision between capillary–gravity solitary waves in shallow water
Summary
The Korteweg–de Vries (KdV) equation [1] is used to represent the asymptotic behavior of weakly nonlinear two-dimensional unidirectional solitary waves in shallow water. Gardner et al [2] have shown the exact solution for the solitary waves with the help of the inverse scattering transform (IST) scheme. One noticed that when the solitary waves come closer, they collide, interchange their position and energies, and obtain their real shapes after the separation process. During the whole process of collision, the solitary waves have extraordinary stable properties. The unique effect in this entire process of collision is the phase shifts. It is believed that the solitary waves with striking colliding properties can only be kept in a conservative system
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