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New Lax pairs of a shallow water wave model of generalized KdV equation type are presented. According to this Lax pair, we constructed a new spectral problem. By using this spectral problem, we constructed Darboux transformation with the help of a gauge transformation. Applying this Darboux transformation, some new exact solutions including double-soliton solution of the shallow water wave model of generalized KdV equation type are obtained. In order to visually show dynamical behaviors of these double soliton solutions, we plot graphs of profiles of them and discuss their dynamical properties.

It is well known that the Lax pair and Darboux transformation can be employed to obtain multisoliton solution of nonlinear evolution equations. Darboux transformations provide us with purely algebraic, powerful method to construct solutions for systems of nonlinear equations. In recent years, more and more researchers used the Lax pair and Darboux transformation to investigate soliton solutions of classical nonlinear wave equations and some new soliton equations which were generated by new spectral problems; see [

In this paper, we will investigate the Lax pairs, Darboux transformation, and double soliton solutions of the following famous shallow water wave model of generalized KdV equation type:

In [

The rest of this paper is organized as follows. In Section

Through a series of tedious computation, we obtain Lax pairs of (

First, we consider the following spectral problems:

Next, we will construct a Darboux Transformation (DT) of the spectral problems (

It means that we have to find a matrix

Suppose

Let

Equations (

Second, we prove the following theory of Darboux transformation for special variable.

Let

Let

Comparing the coefficients of

Finally, by using same way to Theorem

The matrix

Let

Comparing the coefficients of

Substituting (

In this section, we will construct the explicit solutions of the integrable shallow water wave (

Choosing

According to (

Using the Cramer rule to solve the linear algebraic system (

As examples, we will investigate exact solutions of (

The 3-D graphs of profiles of the singular double-soliton solution (

The 3-D graphs of profiles of the singular double-soliton solution (

When

The 2-D graph of profile of the exact soliton solution (

This work is supported by the National Natural Science Foundation of China (no. 11361023), the Natural Science Foundation of Chongqing Normal University (no. 13XLR20), the Scientific Foundation of Education of Yunnan Province (no. 2012C199), and the Program Foundation of Chongqing Innovation Team Project in University under Grant no. KJTD201308.