Integrability, bilinearization, solitons and exact three wave solutions for a forced Korteweg-de Vries equation

Abstract

The Korteweg-de Vries (KdV) equation with a forcing term is considered which demonstrates the mathematical modelling of the physics of a shallow layer of fluid due to external forcing. The characteristic of integrability is investigated via Painlevè analysis and an auto Bäcklund transformation is fabricated via the truncated Painlevè expansion. Using the concept of Bell polynomial approach the bilinear Bäcklund transformation, Lax pair and infinite sequence of conservation laws are constructed. The one-soliton, two-soliton and three-soliton solutions are obtained via simplified Hirota method. We demonstrate the propagation and interaction of soliton solutions graphically. Moreover, a new form of ansatz in the exact three wave method (ETW) along with symbolic computation is applied and some new periodic type of three wave solutions which includes periodic two solitary wave solution, doubly periodic solitary wave solution and breather type of two solitary wave solution are obtained for the forced KdV equation.