KdV hierarchy with time-dependent coefficients: Lax integrability, bilinear Bäcklund transformation and soliton solutions

Starting from the linear problems, we first derive a new Lax integrable Korteweg-de Vries (KdV) hierarchy with time-dependent coefficients. Then a bilinear Bäcklund transformation (BT) of the variable-coefficient KdV (vcKdV) equation contained in the KdV hierarchy is given. Based on the given bilinear BT, one-soliton solution, two-soliton solution and three-soliton solution are obtained. From these obtained soliton solutions, a uniform formula of explicit n-soliton solutions of the vcKdV equation is summarized. It is graphically shown that the dynamical evolutions of such soliton solutions with time-dependent functions of the KdV hierarchy possess time-varying speeds in the process of propagations.