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N-Fold Darboux Transformation for a Nonlinear

Evolution Equation

Yannan Zhao

College of Science, University of Shanghai for Science and Technology, Shanghai, China Email: mathynzhao@yahoo.com.cn

Received June 27, 2012; revised July 27, 2012; accepted August 5, 2012

ABSTRACT

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.

Keywords: Darboux Transformation; Derivative Nonlinear Schrödinger Equation; Explicit Solution

1. Introduction

There are many methods to obtain explicit solutions of nonlinear evolution equations, such as the inverse scat-tering 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 trans-forming 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 di-rectly 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 be-tween 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 rela-tionship between the trivial solution and the general solu-tions. The main idea of this paper comes from the reduc-tion of DT in [20] and the determinant representareduc-tion of Darboux transformation for AKNS system in [21].

In Section 2, from a Lax pair in [22], we deduce sev-eral 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 2

2 2 2

,

x

q r q

U U

r q r q

   

   

    

 

   

 

2

r

(2.1)

and the auxiliary spectral problem

 

 

 

 

11 12

12 11

,

t

V V

V V

V V

 

 

 

    

 

, (2.2)

where

 

3

2

11

1 1 ,

2 x 2

V   q r q qr

 

 

2

 

4 3 2

2 2

2 2

2

2 2

2

12

1 1 1 3

2 .

2 2 x 2 x x 4

V    r  qr  q r qr  rqqr     qr

   

 

(2)

We get a new nonlinear evolution equation

2

2 2 2 2 2 2 2

2

2 2 2 2 2 2 2

1 1 2 1 2

2 2 2

1 1 2 1 2

2 2 2

t xx x x x

x

t xx x x x x

q r q q r q q r r rq qr r q r

r q r q r r q r q rq qr q q r

    

    

 

   

    

   

        

,

.

(2.3)

Letting u r iq  , the above system reduces to a generalized derivative nonlinear SchrÖdinger (GDNS) equation

 

2 2

 

* 2

1 1 1

2 Im 2

2 2 2

t xx x x

x

iuu  u ui u u  u uu    u u

   

4

. (2.4)

If 0, 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 1 2. In this case, from (2.3), we have

2

2 2 2 2

2

2 2 2 2

1 1 1 ,

2 2 4

1 1 1 ,

2 2 4

t xx x x x

t xx x x x

q r q q r r rq qr r q r

r q r q r q rq qr q q r

        

 

       



(2.5)

and when u r iq  , the above system becomes

 

2 * 4

1 Im 1 ,

2 2 4

t xx x x

i

iuuu uu uuu u (2.6) In which u* 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 ob-tained. Also, the relationship between

 

q N1 ,r N1

and

q N r N

   

,

is given. By us- ing of this relationship, if we want to get

q N r N

   

,

, we have to deduce

q i

   

,r i

for . This is very complicated. The purpose of this paper is to im-prove this process and obtain the relationship between the trivial solution

1, 1

i N

  

q 0 , 0r

and the new solution

   

q N r N,

by constructing N-fold Darboux matrix.

3. Darboux Transformation

We first introduce a transformation

T

   (3.1) for the spectral problem (2.1), where T satisfies

0. x

TTU UT  (3.2) Note that U and U have the same form except that q

and r are replaced by q and r, respectively, in (2.1). According to the forms of (2.1) and (2.2), we find that if

 ,

T is a solution with   1,

,

T

 

 

1 1

0 0

1 1 1

0 0

1

, 1

N N

N k

N k

k k

k k

N N k N

k N k

k k

k k

a b

T

b a

 

  

 

 

 

 

 

 

 

 

 

k

(3.3)

where ak and bk

0  k N 1

are functions of x and t.

Let

 

1

   

, 2

T

j j

    

  j

and

 

1

   

, 2

T

j j

    

  j

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

 

2

 

, 1

 

T

j j

    

   j

and

 

2

 

, 1

 

T

j j

    

   j

are two basic solutions of (2.1) and (2.2) with = j

 

j1, 2, , N

. From (3.3), we find that

2 2 1

detT jN  j , (3.4)

  is a solution with  1. Then we suppose that T has the

following form

which means that j and j

a

are roots of det =0T . From (3.1), we find that and satisfy the fol-lowing linear algebraic system k k

(3)

 

 

1 0 1 1 0 1 , 1 , N

N k k N

k j k j j

k

N N k

k N

j k k j

k a b a b    j j                     

(3.5)

where

 

 

 

 

1 2 2 2

1 1 2 1

j j j j

j

j j j j

l l l l           

 (3.6) with l1j and l2j

j1, 2, , N

are constants.

Proposition 3.1 Through the transformation (3.1) and

(3.2),   x U becomes   x U with

2 2

2 2 2

U

q r q

r q r q

   

   

r2

 

 

(3.7)

and

1

1 2,q q 2bN ,r r 2aN1.

     (3.8)

Proof. Let T1T* detT ( means the adjoint

matrix of T) and

*

T

* 11

 

 

12

 

 

21 22

x

f f

T TU T

f f

 

 

 

 

 . (3.9) It is easy to see that f11

 

 and f22

 

 are (2N +

1)th-order polynomials of  and f22

 

  f11

 

 .

Also, f12

 

 and f21

 

 are (2N + 2)th-order

poly-nomials of  andf21

 

  f12

 

 . When

1

j

, together with (2.1) and (3.6), we

obtain a Riccati equation

j N

  

2 2 2

2 2 2

2

. 2

jx j j j

j j j

r q r q

r q r

j

     

   

     

    (3.10)

After calculation, we find that all

are roots of . Hence we have

1 2

j j N

   1

 

0 ,

1, 2

i j

f   i j

*

det

 

x

TTU TT P  , (3.11) where

 

  11 1  11 0   12 2 2  12 1   12 0

2 2 1 0 1 0

12 12 12 11 11

P P P P P

P

P P P P P

                   

and ( )k

, 1, 2, 0,1, 2 are independent of i j

P i jk  .

Then we have

 

x

TTUPT. (3.12) Comparing the coefficients, we have

      2 2 12 1 1 1

11 -1 12 -1

: 1;

: 2 , 2

N N

N N

P

P q b q P r a r

                        

0 1 2

11 -1 11 -1 -2 12 -2 1

-1 12 -1

0 2 2 1

12 -1 12 -1 -1 11 -1

:

0;

.

N

N N N N

N N

N N N

P qa P a b P b

rb P b

P q r ra P a qb P b

                1 N

We find that U P=

 

 , that is (0)

2 2

12 = +

Pq r if

and only if 1 2. 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

+ t

T TV VT =0 together with 1 2 , q q 2bN-1, -1

2 N .

r  r a   t V is transformed into   t V , where V has the same form as V with q and r replaced by q and r, 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 Ut Vx UV VU 0, we find that both

q r,

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

q r,

through the Darboux trans-formation (3.1) and (3.8), where aN1 and bN1 are

determined by (3.5). And u  r iq 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

q r,

is another solution of (2.5). On the other hand, if (q, r) is a solution of (2.5), then u r iq  is a solution of (2.6). So,

u  r iq 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

 

1 2 , i

1 2

j j j j T

j c ej c ej c ej c ej

   

  

     (4.1)

and

 

1 2 , i

1 2

j j j j T

j c ej c ej c ej c ej

   

  

     , (4.2)

where

π

2

4i, 4

j r ej j r x i r tj j

     and c c1j, 2j,rj are constants. Then

11 22

11

11 22

22

,

1, 2, , .

j j

j j

j j j j j j

j

j j j j j j

l l c e l l c e

i

l l c e l l c e

(4)

From the linear algebraic system (3.5), we have

1 1

1 1

2 2

N N

a b

N N

N N

ab

 

 

 

 ,  , (4.4)

w

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

N here

 

 

 

 

 

1 2 1 2

1 1

1 2 1 2

2 2 2 2 2 2 2

1 2 1 2

2 1 2 1 2 1

1 1 1 1 1 1 1

1

1 2 1 2

2 2 2 2 2 2 2

1 2

1 1

1

1 1

N

N N N N

N

N N N N

N

N N N N

N N N N N N N

N N N N N N

N

N N N N

N N

N N N N

      

      

      

      

      

    

   

   

   

   

   

 

 

 

 

 

 

 

 

       

 

 

 

       

N 1 N 2

 

1 N 1

NNN

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

1 1 1 1  1

   

q 0 , 0r

and

q N r N

   

,

solutions.

-1, -1

N N

a b

 

1)-th columns

1 , 2 , , , 1 1, 2 2, ,

T

N N N N N N

N N

        

       

, which is more direct get

For N = 1, we have and universal to

2

1 1

in 2N, tively. Then, according t

respec

o the Theorem 3.1 and above analy-sis, the solution of nonlinear evolution Equation (2.5) is

 

2 N 1,

 

2 N1

q N   br N   a , (4.5)

and the solution of (2.6) is

 

2 N 1 2i N 1

u N   a   b  . (4.6)

In [16], the relationship of

q N

1 ,

 

r N1

and

   

q N r N,

is given. H formation obtained here gi

o

ns-ve

wever, the Darboux tra s the relationship between

0

1

1

a  12 ,  

 1 12

2 1

0

1

b  

 

th

 , (4.7) en

 

1

1

2 2 2 2

2 2

1 11 11 21 21 11 21

2 2

11 21 11 21

1 i c l l e c l l e ,

q

c c l l

 

 

 (4.8)

 

1

1

2 2 2 2

2 2

1 11 11 21 21 11 21

2 2

11 21 11 21

1 c l l e c l l e

r

c c l l

 

 

 . (4.9)

For N = 2, we have

 



 





2 2 2

1 2 1 1

1 2 2

1 2 1 1 1

a

  

  

2 2 2

2 2 1 2

2 2

2

1 1 1 1

,

      

     

(4.10)

1 2 1 2

4   

 



2 2 2 2

2

then

1 2 2 2 1 1 1 2

1 2 2 2

1 2 1 2 1 2 1 2

2 1 1

1 1 4

b        

       

   

     , (4.11)

2 21

2 2 2

1 2 22 11 21c c c l11 l21 l12 l22 e,

   

 

12 22

1 2 3 4

1 2 3

2 i

q      

  

    

  ,

2

2 2

2 2 2

2 1 21 12 22c c c l12 l22 l11 l21 e ,

    

(4.12)

2 41 22

2 2 2

3 1 11 12 22c c c l12 l22 l11 l21 e  ,

 

12 22

1 2 3 4

1 2 3

2

r      

  

   

  , (4.13) 4 2 12 11 21c c c2

l112 l212

l12l22

2e 1 2,

where

24

 

2

2 41

2 2

1 1 2 11 22c c l11 l21 l12 l22 e ,

(5)

 

2

2 4

2 2

2 1 2 12 21c c l11 l21 l12 l22 e ,

      2

2 2

2 2



2 2

21 2

3 1 2 c c c c11 12 21 22 l11 l21 l12 l2 e .

 

2

If we let 1,

, solutions

11 1, 21 2, 12 3, 22 4 1,

cc cc cc cc l1

21 0,12 0, 22 1

lll

q

   

1 , 1r

and

   

q 2 , 2r

[16].

are exactly the same as the so tions in In general, according to [21], we know that the N

Darboux transformation is an action of the n-times re-peated 1-fold Darboux transformation. The solution

lu

-fold

   

q N r N,

essence. The m

are the same as the solution in [16] in atrix T can be expressed by the determi-nant of basic solutions

11 1 T T12

21 22 2

=

N

T

T T

 

(4.14)

where

2 1

11

2 1 2

=

n N

N N

T  

  ,

2 1 12

2 1 2 0

= N

N N

T

  

,

2 1

21

0 =

ˆ

N

2N1 2N

,

T  

 22 2 1

2 1 2 ˆ

n N

=

ˆ N N

T

with

 

n1,n2, , ,1,0,0, ,0

   ,

2N 1

1 2

2 1

ˆ 0, 0, , 0, n , n , , ,1 N

     

    ,

2 1

1 , 2 , , , 1, 2 2, ,

N

T

N N N N N N

N

        

         N N

.

1 ,

2 1

1 1 2 2 1 2

ˆ

, , , , , , ,

N

N N N N N N

N N N

T

        

       

The matrix T is the same as (3.3), hich is also sistent with the Darboux matrix in [21

5. Conclusion

In this paper, for a Lax pair which is not the AKNS sys-tem, we give a N-fold Darboux transformation, coeffi-cients of this matrix can be obtained from a algebraic system and expressed with rank-2N determinants. The Darboux transformation gives the relationship between

w con-

].

   

q 0 , 0r

and

q N r N

   

,

The advantage of this method is that it is more direct and universal to get ex-plicit solutions of (2.5) and (2.6).

Transformations Geometry an

REFERENCES

[1] M. J. Ablowitz and H. Segur, “Solitons and Inverse

Scat-tering Transform (SIAM Studies in Applied Mathematics, No. 4),” Society for Industrial Mathematics, 2000. [2] C. Rogers and W. K. Schief, “Bäcklund and Darboux

d Modern Application in Soliton Theory,” Cambridge University Press, Cambridge, 2002. doi:10.1017/CBO9780511606359

[3] C. H. Gu, H. S. Hu and Z. X. Zhou, “Darboux Transfor-mations in Integrable Systems: Theory and Their Appli-cations to Geo ysics Studies 26),” Springer-Verla

n and

No. 1, 2005, pp. metry (Mathematical Ph

g, New York, 2005.

[4] A. R. Chowdhury, “Painlevé Analysis and Its Applica-tions (Chapman and Hall/ CRC Monographs and Surveys in Pure and Applied Mathematics 105),” Chapma Hall/CRC, Boca Raton, 1999.

[5] A. M. Wazwaz, “Exact Solutions of Compact and Non-compact Structures for the KP-BBM Equation,” Applied Mathematics and Computation, Vol. 169,

700-712. doi:10.1016/j.amc.2004.09.061

[6] J. H. Li, M. Jia and S. Y. Lou, “Kac-Moody-Virasoro Symmetry Algebra and Symmetry Reductions of the Bi-linear Sinh-Gordon Equation in (2+1)-dimensions,” Jour- nal of Physics A: Mathematical and Theoretical, Vol. 40, No. 7, 2007, pp. 1585-1595.

doi:10.1088/1751-8113/40/7/010

[7] S. L. Zhang, S. Y. Lou and C. Z. Qu, “Functional Vari-able Separation for Extended (1+2)-Dimensional Nonlin-ear Wave Equations,” Chinese Physics Letters, Vol. 22, No. 11, 2005, pp. 2731-2734.

doi:10.1088/0256-307X/22/11/001

[8] Gegenhasi, X. B. Hu and H. Y. Wang, “A (2+1)-Dimen-sional Sinh-Gordon Equation and Its Pfaffian Generaliza-tion,” Physics Letters A, Vol. 360, No. 3, 2007, pp. 439-447. doi:10.1016/j.physleta.2006.07.031

[9] X. J. Liu and Y. B. Zeng, “On the tions with Self-consistent Sources,

Ablowitz-Ladik Equa-” Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 30, 2007, pp. 8765-8790. doi:10.1088/1751-8113/40/30/011

[10] S. Y. Lou, M. Jia, X. Y. Tang and F. Huang, “Vortices, Circumfluence, Symmetry Groups,

formations of the (2+1)-Dimensio

and Darboux Trans-nal Euler Equation,” Physical Review E, Vol. 75, No. 5, 2007, Article ID: 056318. doi:10.1103/PhysRevE.75.056318

[11] C. L. Chen, Y. S. Li and J. E. Zhang, “The Multi-Soliton Solutions of the CH-γ Equation,” Science in China Series A, Vol. 51, No. 2, 2008, pp. 314-320.

doi:10.1007/s11425-007-0137-x

[12] L. J. Zhou, “Darboux Transformation for the Nonisospec-tral AKNS System,” Physics Letters A, Vol. 345, No. 4-6, 2005, pp. 314-322. doi:10.1016/j.physleta.2005.07.046 [13] H. C. Hu, “Analytical Positon, Negaton and Complexiton

Solutions for the Coupled Modified KdV System,” Jour-nal of Physics A: Mathematical and Theoretical, Vol. 42, No. 18, 2009, Article ID: 185207.

doi:10.1088/1751-8113/42/18/185207

[14] A. H. Chen and X. M. Li, “Soliton Solutions of the Cou-pled Dispersionless Equation,” Physi

370, No. 3-4, 2007, pp. 281-286.

(6)

o [15] J. Wang, “Darboux Transformation and Soliton Solutions

for the Boiti-Pempinelli-Tu (BPT) Hierarchy,” Journal f Physics A: Mathematical and General, Vol. 38, No. 39, 2005, pp. 8367-8377. doi:10.1088/0305-4470/38/39/005 [16] L. Luo, “Darboux Transformation and Exact Solutions for

A Hierarchy of Nonlinear Evolutio

of Physics A: Mathematical and Theoren Equations,” tical, Vol. 40, No. Journal 15, 2007, pp. 4169-4179.

doi:10.1088/1751-8113/40/15/008

[17] Y.Wang, L. J. Shen and D. L. Du, “Darboux tion and Explicit Solutions for Som

Transforma-e (2+1)-DimTransforma-ensional Equations,” Physics Letters A, Vol. 366, No. 3, 2007, pp. 230-240. doi:10.1016/j.physleta.2007.02.043

[18] X. G. Geng and H. W. Tam, “Darboux Transformation and Soliton Solutions for Generalized Nonlinear Schr- Ödinger Equations,” Journal of the Physical Society of Japan, Vol. 68, No. 5, 1999, pp. 1508-1512.

doi:10.1143/JPSJ.68.1508 [19] X. M. Li and A. H. Chen,

Multi-soliton Solutions of Boussine

“Darboux Transformation and sq-Burgers Equation,”

Physics Letters A, Vol. 342, No. 5-6, 2005, pp. 413-420. doi:10.1016/j.physleta.2005.05.083

[20] H. X. Wu, Y. B. Zeng and T. Y. Fan, “Complexitons of the Modified KdV Equation by Darboux Transforma-tion,” Applied Mathematics and Computation, Vol. 196, No. 2, 2008, pp. 501-510. doi:10.1016/j.amc.2007.06.011

. S. Li, “Determinant [21] J. S. He, L. Zhang, Y. Cheng and Y

Representation of Darboux Transformation for the AKNS System,” Science in China Series A, Vol. 49, No. 12, 2006, pp. 1867-1878. doi:10.1007/s11425-006-2025-1 [22] Z. Y. Yan and H. Q. Zhang, “A Lax Integrable Hierarchy,

N-Hamiltonian Structure, r-Matrix, Finite-Dimensional Liouville Integrable Involutive Systems, and Involutive Solutions,” Chaos Solitons and Fractals, Vol. 13, No. 7, 2002, pp. 1439-1450.

doi:10.1016/S0960-0779(01)00150-3

[23] E. Mjolhus and J. Wyller, “Nonlinear Alfvén Waves in a Finite-Beta Plasma,” Journal of Plasma Physics, Vol. 40, No. 2, 1988, pp. 299-318.

References

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