Numerical investigations of waves interactions from forced Korteweg de Vries equations

Abstract

Soliton generated by the Korteweg de Vries (KdV) equation forms a group of solitons ladder. During full interaction of multi-soliton solutions, three types of peaks were obtained, namely single, flat and double peak. Soliton generated by the forced Korteweg de Vries (fKdV) equation forms uniform solitons trains with equal amplitude. Various aspects of solitons interactions of the fKdV equation for free surface flow over uneven bottom topography have been investigated. Fluid flowing over uneven bottom topography can support wave propagation that generates upstream and downstream nonlinear wavetrains. Such forced nonlinear solitary waves occur naturally in the shallow water near the coastal region. The fKdV equation models the above phenomena in many cases, such as in the transcritical, weakly nonlinear and weakly dispersive region. Numerical method which involves the pseudo-spectral method is used to solve the fKdV equation as it is difficult to obtain the solution analytically, due to the presence of the forcing term and the broken symmetry. A group of uniform solitons having the same amplitude and speed will not collide when the bump size and bump speed are constant. A wave profile with time-dependent transcritical velocity was investigated with a variation of Froude number. As the Froude number changes, two sets of solitary waves travelling upstream were discovered. A set of these solitary waves have nearly uniform amplitude, while another set comprises of solitary waves with variable amplitude, which forms a pairwise and two pairwise interactions pattern in the transcritical region. In the case of multiple bumps, upstream-advancing nonlinear solitary waves which may be generated continuously and interact with each other when the distance between bumps, width and height of bumps were varied. Interesting interaction patterns of the collision between uniform solitons will provide a better understanding of the forcing caused by multiple bumps on water flow at the uneven bottom topography of a shallow water in a rectangular channel.