_{1}

^{*}

Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.

Studying of the nonlinear models in shallow water wave is very important, such as Korteweg-de Vries (KdV) equation [

In this paper, we investigate the Whitham-Broer-Kaup (WBK) system [

where

Many solutions have been obtained for the WBK system (1), such as the analytical solution, the soliton-like solution, the soliton solutions, the periodic solution, the rational solution, and so on [

In this paper, through a proper transformation

the WBK system (1) is transformed into the parameter Levi system

Based on the obtained Lax pair, we construct the N-fold DT of the parameter Levi system (3) and then get the N-fold DT of the WBK system (1). Resorting to the obtained DT, we get new multi-soliton solutions of the WBK system.

The paper is organized as follows. In Section 2, we construct the N-fold DT of the Levi system and the WBK system. In Section 3, DT will be applied to generate explicit solutions of the WBK system (1).

In this section, we first construct the N-fold DT of the parameter Levi system, and then get explicit solutions of the WBK system.

We consider the following spectral problem corresponding to the Levi system (3)

and its auxiliary problem

where

Now we introduce a transformation of (4) and (5)

where

Then the Lax pair (4) and (5) are transformed into

where

In order to make the Lax pair (4) and (5) invariant under the transformation (6), it is necessary to find a matrix

Let the matrix

with

where

Let

with

where the constants

then

From (10), we have

We note that (11) can be written as a linear algebraic system

and

which implies that

where

Proposition 1. Let

Then the matrix

where the transformations from the old potentials

Proof: Let

where

By using (16) and (20), we can prove that

Hence, together with (19), we have

that is

with

where

Substituting (17) into (24)-(26) yields

From (7) and (22), we find that

Remark. When

Let the basic solution

Proposition 2. Suppose

Then the matrix

where

The proof of Proposition 2 is similar with Proposition 1, but it is much more tedious and then we omit the proof for brevity. For the similar proof we can also refer to [

According to Proposition 1 and 2, the Lax pair (4) and (5) is transformed into the Lax pair (8) and (9), then the transformation (6) and (18):

Theorem 1. If

is another solution of the parameter Levi system (3), where

From the transformation (2), we find that

Theorem 2. If

is another solution of the WBK system (1), where

In this section, we take a trivial solution

Substituting

with

According to (12), we get

For simplicity, we discuss the following two cases, i.e.

As

according to (28), we get

Substituting (35) into (31), we obtain the solution of the WBK system (1) as

with

By choosing proper parameters (such as

As

with

With the help of (30), we get

Then we get another solution of the WBK system (1) by using of (31)

with

When we take

This work is supported by Nurture Funds of National Project of University of Shanghai for Science and Technology (no. 14XPQ09).