Abstract
First, by improving some key steps in the homogeneous balance method, a new auto-Bäcklund transformation (BT) to the KdV equation with general variable coefficients is derived. The new auto-BT in this paper does not require the coefficients of the equation to be linearly dependent. Then, based on the new auto-BT in which there is only one quadratic homogeneity equation to be solved, an exact soliton-like solution containing 2-solitary wave is given.
Highlights
The KdV equation with general variable coefficients reads ut + f (t) uux + g (t) uxxx = 0, (1)where f(t) and g(t) are arbitrary analytic functions of t, which was originally proposed in [1]
Much progress has been made in the studies of (1) obtaining its auto-Backlund transformation (BT) and exact solutions [2,3,4,5,6]
One can see that the new auto-BT in this paper does not require the coefficients of the equation to be linearly dependent
Summary
The KdV equation with general variable coefficients reads ut + f (t) uux + g (t) uxxx = 0, (1). Where f(t) and g(t) are arbitrary analytic functions of t, which was originally proposed in [1]. Equation (1) is well known as a model equation describing the propagation of weakly nonlinear and weakly dispersive waves in inhomogeneous media. Much progress has been made in the studies of (1) obtaining its auto-BT and exact solutions [2,3,4,5,6]. In [3], by using the truncate Painleveexpansion [7], Hong and Jung obtained an auto-BT which includes 5 Painleve-Backlund equations to be solved. Wang derived an auto-BT of (1) as follows:
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