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.

1. Introduction

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 [130] and references cited therein. In general, a systematical theory on such Darboux transformation even for matrix spectral problem and the resulting zero curvature equation has a beautiful algebraic structure for associated evolution equations, which tells symmetry algebras of the obtained evolution equations; see [31, 32] and references cited therein. Sometimes, it is found that there are many infinity symmetries from the adopted zero curvature equation.

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: which appeared in [33], where is a small-amplitude parameter. Only dropping the right-hand side of (1) gives BBM equation. Dropping the right-hand side of (1) and replacing the term by the term give KdV equation [34]. Thus, (1) can be seen as a BBM equation extended by retaining higher order terms in an asymptotic expansion in terms of the small-amplitude parameter . Dropping the term from (1) and letting , (1) becomes the celebrated Camassa-Holm equation [33] as follows: where is the fluid velocity in the direction (or equivalent to the height of the water’s free surface above a flat bottom), is a constant related to the critical shallow water wave speed, and subscripts denote partial derivatives. Letting , (1) can be rewritten as which comes from physical considerations via the methodology introduced by Fuchssteiner and Fokas in [35, 36]. The Lax pairs of (3) with are given by [37, 38] as follows: where is an arbitrary constant. It is a pity that the [37] is not a formal publication, and it is only preprint paper, so we cannot know its reality contents whether the authors have obtained soliton solutions of (3) by the Lax pairs (4) and (5). In fact, (3) has been studied by many authors in recent years; see the following brief introductions.

In [39], by using the bifurcation theory of dynamic system, some subsection-function and implicit function solutions such as compactons, solitary waves, smooth periodic waves, and nonsmooth periodic waves with peaks as well as the existence conditions have been presented by Bi. By using the same method, Li and Zhang [40] studied a generalization form of the modified KdV equation, which is more complex than (3). In [40], the existence of solitary wave, kink and antikink wave solutions, and uncountably many smooth and nonsmooth periodic wave solutions are discussed. By using the improved method named integral bifurcation method [41], Rui et al. [42] obtain all kinds of soliton-like or kink-like wave solutions, periodic wave solutions with loop or without loop, smooth compacton-like periodic wave solutions, and nonsmooth periodic cusp wave solutions for (3). In [43], Long and Chen discussed the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave, and compactons were for (3). From the above references, (1) (i.e., (3)) is a very important water wave model.

The rest of this paper is organized as follows. In Section 2, we will derive new Lax pair and Darboux transformation of (1). In Section 3, by using this Darboux transformation, we will investigate soliton solutions of (1) and discuss the dynamic properties of these soliton solutions.

2. Lax Pair and Darboux Transformation of (1)

Through a series of tedious computation, we obtain Lax pairs of (1) as follows: Obviously, the Lax pairs (6) and (7) are different from the Lax pairs (4) and (5) under . They are new Lax pairs which we obtained. By using the new Lax pairs (6) and (7), we will construct a Darboux transformation for obtaining soliton solutions of (1).

First, we consider the following spectral problems: withwhere is a constant, is a spectral parameter, and is a potential function. From compatibility, condition yields a zero curvature equation . Substituting into the zero curvature equation, by a direct calculation, (1) is obtained successfully.

Next, we will construct a Darboux Transformation (DT) of the spectral problems (8). In fact, the DT is actually a gauge transformation of the spectral problems (8). It is required that also satisfies the same form of spectral problems

It means that we have to find a matrix such that the old potential is replaced by the new one .

Suppose where and , and are functions of and .

Let be two basic solutions of (8). From (10), there exist constants , which satisfy Further, (16) can be written as a linear algebraic system That is where and the constants as   are suitably chosen such that determinant of coefficients for (18) is nonzero. Therefore, , and are uniquely determined by (18).

Equations (14) and (15) show that the is a th-order polynomial in , and On the other hand, from (17), we have . Thus we have where is independent of . Equation (21) implies that are roots of .

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

Theorem 1. Let satisfy Then the matrix determined by (11) has the same form as ; that is, where the transformation from the old potential into new one is given by

Proof. Let and It is easy to see that and are th-order polynomials in , and are th-order polynomials in . From (19) and (8), we find Through direct calculation, all are roots of . Together with (21) and (26), we get with where are independent of spectral parameter . Indeed, (28) can be written as
Comparing the coefficients of in (30), we find Substituting (22) into (31) yields From (22), (24), (25), and (31) and noticing in (23), we get Thus . The proof of Theorem 1 is completed.

Finally, by using same way to Theorem 1, we prove that in (12) has the same form as under the transformation (10) and (24); see the following theory and its proof.

Theorem 2. The matrix defined by (12) has the same type as , in which the old potential is mapped into via the same DT (24).

Proof. Let and It is easy to see that and are th-order polynomials in , and are th-order polynomials in . By using (19) and (8), we obtain Through direct calculation, all are roots of . Together with (21) and (34), we get with where are independent of spectral parameter . Equation (37) can be written as
Comparing the coefficients of and in (38) leads to
Substituting (22), (24), and (25) into (40) to (43), we can get On the other hand, from (24), (25), and (33), we have From (44) and (45), obviously we have Thus , and this completes the proof of Theorem 2.

3. Exact Soliton Solution of (1) and Its Dynamic Properties

In this section, we will construct the explicit solutions of the integrable shallow water wave (1) by using the Darboux transformation (24).

Choosing ( is an arbitrary constant) as a seed solution of (1), and substituting into the spectral problems (8). Then we get two basic solutions of (8) as follows: with where is a nonzero constant and .

According to (19), we have where are nonzero constants.

Using the Cramer rule to solve the linear algebraic system (18), we obtain where

As examples, we will investigate exact solutions of (1) in two simple cases and . When , solving the linear algebraic system (18) leads to Substituting (49) and (52) into (24), a singular double-soliton solution of (1) is obtained as follows: where By using program of computer, it is easy to verify that the solution (53) satisfies (1), and this shows that the Darboux transformation (24) which we obtained is correct. In order to show the properties of the above singular double-soliton solutions visually, as an example, we plot the 3-D graphs of solution (53) for some fixed parameters, which are shown in Figures 1 and 2.

When , using the Cramer rule to solve the linear algebraic system (18), we obtain with where are given by (49). From (24), an explicit solution of (1) is obtained by the following: where is given by (55). Equation (57) is a very complex solution, and it is not soliton solution. In order to show the properties of solution (57), under the fixed parameters , , , , , , , , , , , , , we plot its 2-D profile, which is shown in Figure 3.


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.