There are many methods to obtain explicit solutions of nonlinear evolution equations, such as the inverse scattering transformation (IST) [1], Bäcklund transformation [2], Darboux transformation [3], Painlevé analysis method [4], etc. [5-8]. Among these methods, DT is a useful method which is a special gauge transformation transforming a linear problem into itself. Many different forms of DTs have been considered in [9-19]. Generally, there are two kinds for the DTs of Lax pairs. One is to give 1-fold form Darboux matrix and obtain the N-th solution by iterating N times [14-16]. The other is to directly construct N-fold form Darboux matrix and obtain the N-th solution without iteration [17-19].

In this paper, according to the form of a Lax pair and the properties of solutions, we consider the relations between the above two kinds of DTs by considering the Lax pair given in [16]. We construct the N-fold DT for the Lax pair, which simplifies the complicated process of iterating 1-fold form Darboux matrix and gives the relationship between the trivial solution and the general solutions. The main idea of this paper comes from the reduction of DT in [20] and the determinant representation of Darboux transformation for AKNS system in [21].

In Section 2, from a Lax pair in [22], we deduce several nonlinear evolution equations. For a special case, we give the N-fold DT. In Section 3, we give the N-fold Darboux matrix and the relationship between new and old potentials. In Section 4, we obtain exact solutions of the nonlinear evolution equations and discuss the properties of these solutions. In Section 5, we make our conclusion.

We consider the isospectral problem introduced in [22]

and the auxiliary spectral problem

where

By using of the zero curvature equation

We get a new nonlinear evolution equation

Letting, the above system reduces to a generalized derivative nonlinear SchrÖdinger (GDNS) equation

If, the above equation reduces to the derivative nonlinear SchrÖdinger (DNS) equation which describes the propagation of circular polarized nonlinear Alfvén waves in plasmas [23].

We consider the Darboux transformation of the Lax pair (2.1) and (2.2). In this paper, we find the Darboux transformation for the case of. In this case, from

(2.3), we have

and when, the above system becomes

In which means the conjugate of u.

In [16], 1-fold Darboux matrix has been given and by applying it N times, a series of explicit solutions are obtained. Also, the relationship between

and is given. By using of this relationship, if we want to get, we have to deduce for. This is very complicated. The purpose of this paper is to improve this process and obtain the relationship between the trivial solution and the new solution by constructing N-fold Darboux matrix.

We first introduce a transformation

for the spectral problem (2.1), where T satisfies

Note that and U have the same form except that q and r are replaced by and, respectively, in (2.1).

According to the forms of (2.1) and (2.2), we find that if is a solution with, is a solution with. Then we suppose that T has the following form

where and are functions of x and t.

Let

and

be two basic solutions of (2.1) and (2.2) with. Then

and

are two basic solutions of (2.1) and (2.2) with. From (3.3), we find that

which means that and are roots of.

From (3.1), we find that and satisfy the following linear algebraic system

where

with and are constants.

Proposition 3.1 Through the transformation (3.1) and (3.2), becomes with

and

Proof. Let (means the adjoint matrix of T) and

It is easy to see that and are (2N + 1)th-order polynomials of and. Also, and are (2N + 2)th-order polynomials of and When

together with (2.1) and (3.6), we obtain a Riccati equation

After calculation, we find that all are roots of. Hence we have

where

and are independent of. Then we have

Comparing the coefficients, we have

We find that, that is if and only if. The proof is complete.

Remark 3.1 The proof of Proposition 3.1 is similar to that in [18]. Due to the property of basic solutions of the Lax pair (2.1) and (2.2), the proof here is more tricky.

Proposition 3.2 From the transformation (3.1) and together with, is transformed into, where has the same form as V with q and r replaced by and, respectively.

Remark 3.2 The proof of Proposition 3.2 is similar to Proposition 3.1 and we omit it here for brevity.

According to Proposition 3.1 and 3.2, from the zero curvature equation, we find that both and (q, r) satisfy (2.5). The transformation (3.1) and (3.8) is called the Darboux transformation of (2.5). Then we have the following theorem.

Theorem 3.1 The solution (q, r) of (2.5) are mapped into the new solution through the Darboux transformation (3.1) and (3.8), where and are determined by (3.5). And is a new solution of (2.6).

Proof. On one hand, according to the Proposition 3.1 and 3.2, together with the transformation (3.1), we know that (q, r) is a solution of (2.5), and is another solution of (2.5). On the other hand, if (q, r) is a solution of (2.5), then is a solution of (2.6). So, is a new solution of (2.6).

In this section, we apply the Darboux transformation (3.1) and (3.8) to get explicit solutions of (2.5).

To compare with the solutions obtained in [16], we start from the same trivial solution (q[0], r[0]) = (0,0) and select basic solutions

and

where and are constants. Then

From the linear algebraic system (3.5), we have

where

and are obtained by replacing 1-st and (N + 1)-th columns with

in respectively.

Then, according to the Theorem 3.1 and above analysis, the solution of nonlinear evolution Equation (2.5) is

and the solution of (2.6) is

In [16], the relationship of and is given. However, the Darboux transformation obtained here gives the relationship between

and, which is more direct and universal to get solutions.

For N = 1, we have

, , (4.7)

then

For N = 2, we have

then

where

If we let

, solutions and

are exactly the same as the solutions in [16].

In general, according to [21], we know that the N-fold Darboux transformation is an action of the n-times repeated 1-fold Darboux transformation. The solution are the same as the solution in [16] in essence. The matrix T can be expressed by the determinant of basic solutions

where

, ,

,

with

,

,

The matrix T is the same as (3.3), which is also consistent with the Darboux matrix in [21].

In this paper, for a Lax pair which is not the AKNS system, we give a N-fold Darboux transformation, coefficients of this matrix can be obtained from a algebraic system and expressed with rank-2N determinants. The Darboux transformation gives the relationship between and The advantage of this method is that it is more direct and universal to get explicit solutions of (2.5) and (2.6).