1. Introduction
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.
2. Soliton Equations
We consider the isospectral problem introduced in [22]
(2.1)
and the auxiliary spectral problem
(2.2)
where
By using of the zero curvature equation
We get a new nonlinear evolution equation
(2.3)
Letting
, the above system reduces to a generalized derivative nonlinear SchrÖdinger (GDNS) equation
(2.4)
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
(2.5)
and when
, the above system becomes
(2.6)
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.
3. Darboux Transformation
We first introduce a transformation
(3.1)
for the spectral problem (2.1), where T satisfies
(3.2)
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
(3.3)
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
(3.4)
which means that 
and 
are roots of
.
From (3.1), we find that 
and 
satisfy the following linear algebraic system
(3.5)
where
(3.6)
with 
and 
are constants.
Proposition 3.1 Through the transformation (3.1) and (3.2), 
becomes 
with
(3.7)
and
(3.8)
Proof. Let 
(
means the adjoint matrix of T) and
(3.9)
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
(3.10)
After calculation, we find that all 
are roots of
. Hence we have
(3.11)
where
and 
are independent of
. Then we have
(3.12)
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).
4. Explicit Solutions
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
(4.1)
and
, (4.2)
where 
and 
are constants. Then
(4.3)
From the linear algebraic system (3.5), we have
, (4.4)
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
(4.5)
and the solution of (2.6) is
. (4.6)
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
(4.8)
. (4.9)
For N = 2, we have
(4.10)
(4.11)
then
(4.12)
(4.13)
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
(4.14)
where
, 
,
, 
with
,
,
The matrix T is the same as (3.3), which is also consistent with the Darboux matrix in [21].
5. Conclusion
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).