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In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.

There are many methods to obtain explicit solutions of nonlinear evolution equations, such as the inverse scattering transformation (IST) [

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 [

In Section 2, from a Lax pair in [

We consider the isospectral problem introduced in [

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 [

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 [

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 [

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 [

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 [

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 [

In general, according to [

where

, ,

,

with

,

,

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

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).