Backlund Transformation and N-soliton-like Solutions to the Combined KdV-Burgers Equation with Variable Coefficients

Backlund Transformation and N-soliton-like Solutions to the Combined KdV-Burgers Equation with Variable Coefficients ~ In the soliton theory, a remarkable aspect is that the nonlinear evolution equations governing them are exactly solvable[1]-[4]. Especially, it has been shown by analytical investigations that the existence of N-soliton solution describing multiple collision solutions is a common algebraic feature of these nonlinear evolution equations. There exist several direct methods to construct N-soliton solution, among them the Backlund transformations method, The advantage of this method is that the N-soliton solution is represented by a purely algebraic process. The method provides a powerful tool in searching for soliton solutions of the partial differential equations(PDEs), including the PDEs with constant coefficients and variable coefficients, some of its application can be referred to in [5]-[9]. In this paper, we consider the combined KdV-Burgers equation with variable coefficients in the form
ut + α(t)uux − β(t)u2ux + µ(t)uxx + δ(t)uxxx = 0
where α(t),β(t),µ(t)and δ(t) are arbitrary functions of t. When β(t)=0,α(t),µ(t)and δ(t) are constants, Eq.(1) turns to KdV-Burgers equation,which mainly describes the flow of air bladder in liquid and the flow of liquid within ballistic trajectory canal. When µ(t) = 0,α(t),β(t),δ(t) are constants, Eq.(1) turns to KdV-MKdV equation,which is widely used in solid physics, atom physics and quanta field theory etc. Some of the research can be referred to in [10]-[14]. When β(t) = µ(t)=0 it is the case of [15]-[17]. In this work we would like to use the HBP to derive a BT of Eq.(1), via the BT rational fraction solutions and N-soliton- like solutions of Eq.(1) can be completely obtained. As an illustrative example, double-soliton solution representing the interaction of two solitary wave solutions and three-soliton solution of the combined KdV- Burgers equation are given. To our knowledge, this result of combined KdV-Burgers equation with variable coefficients has never been seen before in the literature. In addition, the soliton solutions possess a variable propagating speed that depends upon the coefficients of the equation.

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