Lax pair, conservation laws and N-soliton solutions for the extended Korteweg-de Vries equations in fluids

Abstract

Under investigation in this paper are two extended Korteweg-de Vries (eKdV) equations in fluids with the second-order nonlinear and dispersive terms. Based on the Ablowitz-Kaup-Newell-Segur system, the Lax pair and infinitely many conservation laws are derived. By virtue of the Hirota method and symbolic computation, the bilinear forms and N-soliton solutions for the two eKdV equations are obtained, respectively. Relevant propagation properties and interaction behaviors of the solitons are illustrated graphically. The collisions for the η profile are proved to be elastic through the asymptotic analysis. Types of collisions (head-on or overtaking collisions) can be controlled when we adjust the sign of the velocity v. Velocities of solitons are related to c 4 and α during the collisions. Moreover, there is not a direct proportion relationship between the velocity v and amplitude a during the collisions. On the one hand, the soliton with the larger amplitude travels faster and catches up with the smaller one. On the other hand, the soliton with the smaller amplitude travels faster and catches up with the larger one.