Soliton collisions and integrable aspects of the fifth-order Korteweg-de Vries equation for shallow water with surface tension

Abstract

The fifth-order Korteweg-de Vries (KdV) equation works as a model for the shallow water waves with surface tension. Through symbolic computation, binary Bell-polynomial approach and auxiliary independent variable, the bilinear forms, N -soliton solutions, two different Bell-polynomial-type Backlund transformations, Lax pair and infinite conservation laws are obtained. Characteristic-line method is applied to discuss the effects of linear wave speed c as well as length scales τ and γ on the soliton amplitudes and velocities. Increase of τ , c and γ can lead to the increase of the soliton velocity. Soliton amplitude increases with the increase of τ . The larger-amplitude soliton is seen to move faster and then to overtake the smaller one. After the collision, the solitons keep their original shapes and velocities invariant except for the phase shift. Graphic analysis on the two and three-soliton solutions indicates that the overtaking collisions between/among the solitons are all elastic.