A New 2 + 1-Dimensional Integrable Variable Coefficient Toda Equation

In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coefficient Toda equation. The compatibility condition is given, which insures that the new Toda equation is integrable. To further analyze the character of the Toda equation, we derive one soliton solution of the obtained Toda equation by using separation of variables.


Introduction
Integrable variable coefficient equations describe the real world in many fields of physical and engineering sciences. Many researchers are devoted to discussing these equations by utilizing different methods ref. [1]- [6]. In ref. [7] [8], Dai and Jeffrey extended the dressing method to a generalized version for solving nonlinear evolution equations associated with matrix spectral problems and variable coefficient cases, in which a key is that variable coefficient dressing operators are transformed to different variable coefficient ones. By using the generalization, we have studied integrable variable coefficient coupled Hirota equation in ref. [9]. In ref. [10] [11], integrable variable coefficient Manakov model and cylindrical NLS equation are discussed in detailed, respectively. In ref. [12], we developed the generalized dressing method to the discrete system and an integrable variable coefficient Toda equation is researched. Recently, the dressing method is extended to a matrix Lax pair for Camassa-Holm equation in ref. [13], in which interactions between soliton and cuspon solutions of the system are studied. The dressing method as nonlinear superposition in Sigma models has been researched by Dimitrios Katsinis et al. in ref. [14]. Multi-lump solutions of KP equation with integrable boundary are discussed in ref. [15] by using the generalized dressing method. Nabelek et al. in ref. [16] studied Kaup-Broer system and derived its solutions.
In the present paper, we extend the generalized dressing method to discrete operators similar to ref. [12]. Through direct calculations, we derive a new integrable variable coefficient Toda equation where, the coefficient is related to n, is an extension of the well known two-dimensional Toda equation. We will construct one soliton solution of (1.1).
The present paper is organized as follows. In Section 2, we obtain a new integrable variable coefficient Toda equation based on the generalized dressing method. In Section 3, as an application, we derive one soliton solution of (1.1) by utilizing the separation of variables.

Integrable Variable Coefficient Toda Equation
In this section, we first summarize the variable coefficient version of the dressing method. We extend the generalized version of the dressing method to discrete systems and derive different integrable cylindrical Toda lattice equations by choosing different operators. First, we consider three linear differential difference operators ref. [12] ( ) ( ) The discrete Gelfand-Levitan-Marchenko (GLM) equation can be obtained from (2.2), which reads in ref. [12] where E is the shift operator of the discrete variable n, defined by , k Z ∈ , t and y are continuous variables. Acting on function n ϕ on (2.6) and with aid of (2.8), which is reduced to  The following theorem in ref. [7] is an extension of original dressing method, which can yield a wide range of integrable variable-coefficient nonlinear evolution equations.

Explicit Solution of Integrable Variable Coefficient Toda Equation
In this section, we shall use the generalized dressing method to construct explicit solutions of the variable coefficient Toda Equation (  In what follows, we will obtain one soliton solution of (2.21). First, we give separation of variables solutions for