Darboux Transformation and N-Soliton Solution for the CoupledModified Nonlinear Schrodinger EquationsHai-Qiang Zhang

College of Science, P. O. Box 253, University of Shanghai for Science and Technology,Shanghai 200093, China

Reprint requests to H. Z.; E-mail: zhanghqbupt@yahoo.com.cn

Z. Naturforsch. / DOI: 10.5560/ZNA.2012-0084Received May 4, 2012 / revised August 14, 2012 / published online November 14, 2012

The pulse propagation in the picosecond or femtosecond regime of birefringent optical fibers isgoverned by the coupled mixed derivative nonlinear Schrodinger (CMDNLS) equations. A new typeof Lax pair associated with such coupled equations is derived from the Wadati–Konno–Ichikawaspectral problem. The Darboux transformation method is applied to this integrable model, and theN-times iteration formula of the Darboux transformation is presented in terms of the compact de-terminant representation. Starting from the zero potential, the bright vector N-soliton solution ofCMDNLS equations is expressed as a compact determinant by N complex eigenvalues and N lin-early independent eigenfunctions. The collision mechanisms in two components shows that brightvector solitons can exhibit the standard elastic and inelastic collisions. Such energy-exchange colli-sion behaviours have potential applications in the construction of logical gates, the design of fiberdirectional couplers, and quantum information processors.

Key words: Coupled Mixed Derivative Nonlinear Schrodinger Equations; Vector Soliton; SolitonCollision; Darboux Transformation.

PACS numbers: 05.45.Yv; 42.65.Tg; 42.81.Dp

1. Introduction

The coupled mixed derivative nonlinear Schro-dinger equations in the dimensionless form

i q j t +q j xx + µ

(2

∑k=1

|qk|2)

q j

+ iγ

[(2

∑k=1

|qk|2)

q j

]x

= 0 , ( j = 1,2)

(1)

describe the propagation of short pulses in birefrin-gent optical fibers both in picosecond and femtosec-ond regions [1 – 5], where q j = q j(x, t) ( j = 1,2) isthe slowly varying complex envelope for polariza-tions, x and t appended to q j denote partial differ-entiations, the parameters µ and γ are real constantsas the nonlinearity and derivative nonlinearity coeffi-cients. For (1), the case γ = 0 is known as the couplednonlinear Schrodinger (CNLS) equations [6, 7], andµ = 0 is the coupled derivative nonlinear Schrodinger

(CDNLS) equations governing the polarized Alfvenwaves in plasma physics [8]. Thus, (1) could be re-garded as a hybrid of CNLS and CDNLS equations.From the integrable viewpoint for nonlinear evolu-tion equations (NLEEs), (1) possess infinitely manylocal conservation laws [5, 9], Lax pairs [2, 5], andbilinear representations [3 – 5]. In addition, the exactbright N-soliton, dark and antidark soliton solutionshave been presented by employing Hirota’s bilinearmethod [3 – 5, 10].

In soliton theory, the Darboux transformationmethod has been a very effective tool to con-struct the exact analytical solutions of integrableNLEEs [11 – 16], especially the N-soliton solutions.The purpose of the transformation is to produce newsolutions by solving the linear equations with a triv-ial solution. The efficiency of the Darboux trans-formation is due to the fact that the iterative algo-rithm is purely algebraic and can be implementedon the symbolic computation system. With a succes-sive application of the Darboux transformation, the N-

© 2012 Verlag der Zeitschrift fur Naturforschung, Tubingen · http://znaturforsch.com

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

soliton solutions of NLEEs can be presented in a sim-ple and compact form, such as in the form of theWronskian determinant or Vandermonde-like determi-nant. The N-soliton solutions in terms of the deter-minant representations have been revealed for someintegrable multi-component equations, such as themulti-component modified Korteweg–de Vries equa-tions [17], multi-component nonlinear Schrodinger(NLS) equations [18, 19], and two-component deriva-tive NLS equations [20, 21]. However, when the Dar-boux transformation is applied to the multi-componentsystems, the reduction problem between or amongoriginal potentials remains yet as a matter of technicaldifficulties.

The aim of the present work is to construct theDarboux transformation for (1) and to present the N-soliton solution in terms of the determinant represen-tation. On the basis of the N-soliton solution, we willfind that the bright vector solitons in two componentsfor (1) can exhibit the standard elastic and inelastic col-lisions. In inelastic collision, we will identify that eachbright vector soliton can undergo the partial or com-plete energy switching.

The structure of this paper will be arranged as fol-lows. In Section 2, we will derive a new type of the Laxpair of (1) by means of the Wadati–Konno–Ichikawa(WKI) system. Based on the new Lax pair, we will ap-ply the Darboux transformation method to (1). In Sec-tion 3, we will present the N-times iterative potentialformula of (1). In Section 4, starting the zero seed so-lution, we will obtain the bright vector N-soliton solu-tion of (1). Moreover, we will discuss energy-exchangecollision behaviours between or among bright vectorsolitons in two components. Our conclusions will begiven in Section 5.

2. Lax Pair and Darboux Transformation

2.1. Lax Pair

The Lax pair associated with NLEEs is an essen-tial prerequisite for the construction of the Darbouxtransformation. As we know, it is possible that an inte-grable NLEE can admit several Lax pairs. Authors of[2, 5] have derived two kinds of Lax pairs associatedwith (1). In this subsection, based on the WKI spectralproblem [12], we will present another new type of theLax pair for (1).

Following the procedure generalizing the 2×2 WKIscheme to the 3 × 3 case, the Lax pair associatedwith (1) can be written as

Ψx = UΨ =(λ

2U0 +λU1 +U2)Ψ , (2)

Ψt = VΨ =(λ

4V0 +λ3V1 +λ

2V2 +λV3 +V4)Ψ , (3)

whereΨ = (ψ1,ψ2,ψ3)T (the superscript T denotes thevector transpose) is the vector eigenfunction, λ is theeigenvalue parameter, U j and Vk (0 6 j 6 2; 0 6 k 6 4)are all 3×3 matrices to be determined. It is straighfor-ward matter to check that the integrability conditionof (2) and (3),

Ut −Vx +[U,V ] = 0 , (4)

is equivalent to (1) if the matrices U j and Vk (0 6 j 62; 0 6 k 6 4) take the form

U0 =16γ

−2i 0 00 i 00 0 i

,

U1 =12

0 q1 q2−q∗1 0 0−q∗2 0 0

,

U2 =µ

3γ

2i 0 00 −i 00 0 −i

,

(5)

V0 =1

12γ2

−2i 0 00 i 00 0 i

,

V1 =14γ

0 q1 q2−q∗1 0 0−q∗2 0 0

,

V4 =µ2

3γ2

−2i 0 00 i 00 0 i

,

(6)

V2 =µ

3γ2

2i 0 00 −i 00 0 −i

+

i4

|q1|2 + |q2|2 0 00 −|q1|2 −q∗1q2

0 −q1q∗2 −|q2|2

,

(7)

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

V3 =γ

2

0 −q1(|q1|2 + |q2|2)q∗1(|q1|2 + |q2|2) 0q∗2(|q1|2 + |q2|2) 0

−q2(|q1|2 + |q2|2)00

− µ

2γ

0 q1 q2−q∗1 0 0−q∗2 0 0

+i2

0 q1x q2x

q∗1x 0 0q∗2x 0 0

,

(8)

where the asterisk denotes the complex conjugate.

Remark. The above Lax pair (2) and (3) associatedwith (1) is different from those in [2, 5]. Comparedwith the results in [2, 5], the main difference is thatthe traces of matrices U j and Vk (0 6 j 6 2; 0 6 k 6 4)in Lax pair (2) and (3) are all zero. On the basis of thismatrix property, we will construct the Darboux trans-formation for (1).

2.2. Darboux Transformation

The Darboux transformation is a special gaugetransformation which keeps the Lax pair invariant. Inview of the feature that the space part (2) is quadraticabout the spectral parameter λ , we take the Darbouxtransformation for (2) and (3) as the form

Ψ = DΨ =2

∑j=0

λ2− j

a1, j b1, j b2, j

c1, j a2, j d1, j

c2, j d2, j a3, j

Ψ , (9)

which is in terms of the second-order polynomial of λ ,where ak, j, b`, j, c`, j, and d`, j (k = 1,2,3; ` = 1,2; 0 6j 6 2) are all functions of x and t to be determined.

In order to keep the invariance of Lax pair (2)and (3) under the Darboux transformation (9), it re-quires that Ψ also satisfies the same linear eigenvalueproblem (2) and (3) with U j (0 6 j 6 2) and Vk (0 6k 6 4), respectively, replaced by U j and Vk. Thus, theoriginal potentials q1 and q2 are transformed into newpotentials q1 and q2, and the Darboux matrix D is re-quired to satisfy

Dx = UD−DU, Dt = V D−DV . (10)

With use of the Darboux matrix (9), from (10) and bycomparing the coefficients of λ , we can directly com-pute

b1,0 = b2,0 = c1,0 = c2,0 = 0 . (11)

After simplifying the rest equations about each coeffi-cient of λ , we can take

b1,2 = b2,2 = c1,2 = c2,2 = d1,2 = d2,2 = 0 , (12)

a1,1 = a2,1 = d1,1 = d2,1 = d3,1 = 0 , (13)

a1,2 = a2,2 = a3,2 =−1 , (14)

q1 = q1 +2iµ

γb1,1−2b1,1x ,

q2 = q2 +2iµ

γb2,1−2b2,1x ,

(15)

q∗1 = q∗1 +2iµ

γc1,1 +2c1,1x ,

q∗2 = q∗2 +2iµ

γc2,1 +2c2,1x .

(16)

Thus, the Darboux matrix D can be expressed as:

D = λ2

a1,0 0 00 a2,0 d1,00 d2,0 a3,0

+λ

0 b1,1 b2,1c1,1 0 0c2,1 0 0

+

−1 0 00 −1 00 0 −1

.

(17)

Next, our work is to determine the rest entries ak,0,b`,1, c`,1, and d`,0 (k = 1,2,3; ` = 1,2) in D. From theknowledge about the kernel of the Darboux transfor-mation, with Ψ1 = (ψ1,ψ2,ψ3)T as a solution of Laxpair (2) and (3) for λ = λ1, the undermined entries canbe expressed in terms of the eigenvalue and eigenfunc-tion as

a1,0 =1

∆1

(λ∗1 |ψ1|2 +λ1|ψ2|2 +λ1|ψ3|2

), (18)

a2,0 =1

∆2

(|λ1|2|ψ1|2 +λ

2∗1 |ψ2|2 +λ

21 |ψ3|2

), (19)

a3,0 =1

∆2

(|λ1|2|ψ1|2 +λ

21 |ψ2|2 +λ

2∗1 |ψ3|2

), (20)

d1,0 =1

∆2

(λ

2∗1 −λ

21

)ψ2ψ

∗3 ,

d2,0 =1

∆2

(λ

2∗1 −λ

21

)ψ3ψ

∗2 ,

(21)

b1,1 =1

∆1

(λ

2∗1 −λ

21

)ψ1ψ

∗2 ,

b2,1 =1

∆1

(λ

2∗1 −λ

21

)ψ1ψ

∗3 ,

(22)

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

c1,1 =1

∆1

(λ

2∗1 −λ

21

)ψ2ψ

∗1 ,

c2,1 =1

∆1

(λ

2∗1 −λ

21

)ψ3ψ

∗1 ,

(23)

where

∆1 = |λ1|2(λ1|ψ1|2 +λ

∗1 |ψ2|2 +λ

∗1 |ψ3|2

),

∆2 = |λ1|2(λ

2∗1 |ψ1|2 + |λ1|2|ψ2|2 + |λ1|2|ψ3|2

).

It is easy to see that (15) and (16) are satisfied if

b1,1 =−c∗1,1, b2,1 =−c∗2,1. (24)

Making use of (22) and (23), we can directly proof theconstraints (24).

To this state, we have finished the construction ofthe Darboux transformation for (1). Under the transfor-mation (17), the linear spectral problem (2) and (3) isinvariant. At the same time, the new solution (q1, q2)can be obtained by choosing a special eigenvalue λ1and corresponding eigenfunctions (ψ1,ψ2,ψ3) froma seed solution. Therefore, we come to the followingconclusion:

Assume that Ψ1 = (ψ1,ψ2,ψ3)T is a solution of Laxpair (2) and (3) with λ = λ1. The Darboux transforma-tion (Ψ ,q1,q2)→ (Ψ , q1, q2) of (1) is constructed byTransformation (17) with entries defined in (18) – (23).The potential transformation between the new andoriginal ones is given by

q1 = q1 +2(λ

2∗1 −λ

21

)·[

iµγ∆1

ψ1ψ∗2 −(

1∆1

ψ1ψ∗2

)x

],

(25)

q2 = q2 +2(λ

2∗1 −λ

21

)·[

iµγ∆1

ψ1ψ∗3 −(

1∆1

ψ1ψ∗3

)x

].

(26)

3. N-Times Iteration of the DarbouxTransformation

The Darboux matrix D in (9) discussed above isa second-order polynomial in λ . By applying the Dar-boux transformation successively, we can construct theN-times iteration of the Darboux transformation. TheN-times iteration of the Darboux matrix is a 2N-order

polynomial in λ :

Dn(x, t,λ ) =2N

∑k=0

Γkλk (27)

=N

∑n=1

a1,2nλ 2n b1,2n−1λ 2n−1 b2,2n−1λ 2n−1

c1,2n−1λ 2n−1 a2,2nλ 2n d1,2nλ 2n

c2,2n−1λ 2n−1 d2,2nλ 2n a3,2nλ 2n

+(−1)NI,

where I is the 3×3 identity matrix and Γk (0 6 k 6 2N)the coefficient matrix of λ ; Γ2n−1 and Γ2n (0 6 n 6 N)have the following structures:

Γ0 = (−1)NI ,

Γ2n−1 =

0 b1,2n−1 b2,2n−1c1,2n−1 0 0c2,2n−1 0 0

,

Γ2n =

a1,2n 0 00 a2,2n d1,2n

0 d2,2n a3,2n

.

(28)

Let us take a set of linearly independent solutionsΨk = (ψ1,k,ψ2,k,ψ3,k)T of (2) and (3) with (q1,q2) fordifferent spectral parameters λ = λk (1 6 k 6 N), i. e.,Ψk = (ψ1,k,ψ2,k,ψ3,k)T (1 6 k 6 N) satisfy the linearequations (2) and (3):

[∂x−U(λ = λk)]Ψk = 0 ,

[∂t −V (λ = λk)]Ψk = 0 .(29)

We also introduce a set of orthogonal vectors

Φ(1)k =

(−ψ

∗2,k,ψ

∗1,k,0

)T

Φ(2)k =

(−ψ

∗3,k,0,ψ∗1,k

)T,

(30)

which satisfy the orthogonality condition⟨Ψk|Φ

(`)k

⟩= Ψ

†k Φ

(`)k = 0 ,

(1 6 k 6 N; ` = 1,2) ,(31)

where the dagger † signifies the Hermitian conjugate.From the knowledge of the Darboux transformation,the following relations hold:

Dn(x, t,λ )|λ=λkΨk = 0 ,

Dn(x, t,λ )|λ=λ ∗kΦ

(`)k = 0 ,

(1 6 k 6 N; ` = 1,2) ,

(32)

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

which can be rewritten in a matrix form

(Γ1,Γ2, · · · ,Γ2n)Wn =−B, (33)

where

B = Γ0

(Ψ1,Ψ2, · · · ,ΨN ,Φ

(1)1 ,Φ

(1)2 , · · · ,Φ (1)

N ,Φ(2)1 ,Φ

(2)2 , · · · ,Φ (2)

N

),

Wn =

λ1 Ψ1 · · · λN ΨN λ ∗1 Φ

(1)1 · · · λ ∗N Φ

(1)N λ ∗1 Φ

(2)1 · · · λ ∗N Φ

(2)N

λ 21 Ψ1 · · · λ 2

N ΨN λ 2∗1 Φ

(1)1 · · · λ 2∗

N Φ(1)N λ 2∗

1 Φ(2)1 · · · λ 2∗

N Φ(2)N

.

.

. · · ·...

.

.

. · · ·...

.

.

. · · ·...

λ 2N1 Ψ1 · · · λ 2N

N ΨN λ 2N∗1 Φ

(1)1 · · · λ 2N∗

N Φ(1)N λ 2N∗

1 Φ(2)1 · · · λ 2N∗

N Φ(2)N

.

Substituting (28) into linear algebraic (33), we canobtain the expressions of a j,2n,d`,2n,b`,2n−1,c`,2n−1( j = 1,2,3; ` = 1,2; 1 6 n 6 N) by use of Cramer’s

rule. Therefore, the potential formula of N-times itera-tive Darboux transformation for (1) is given by

q1[N] = q1 +(−1)N+1[

2iµγ

b1,1−2b1,1x

]= q1 +(−1)N+1

[2iµ

γ

GH−2

(GH

)x

],

(34)

q2[N] = q2 +(−1)N+1[

2iµγ

b2,1−2b2,1x

]= q2 +(−1)N+1

[2iµ

γ

FH−2

(FH

)x

],

(35)

with

H =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

λ 21 ψ1,1 λ1ψ2,1 λ1ψ3,1 λ 4

1 ψ1,1 λ 31 ψ2,1 λ 3

1 ψ3,1 · · · λ 2N1 ψ1,1 λ

2N−11 ψ2,1 λ

2N−11 ψ3,1

−λ 2∗1 ψ∗2,1 λ ∗1 ψ∗1,1 0 −λ 4∗

1 ψ∗2,1 λ 3∗1 ψ∗1,1 0 · · · −λ 2N∗

1 ψ∗2,1 λ2N−1∗1 ψ∗1,1 0

−λ 2∗1 ψ∗3,1 0 λ ∗1 ψ∗1,1 −λ 4∗

1 ψ∗3,1 0 λ 3∗1 ψ∗1,1 · · · −λ 2N∗

1 ψ∗3,1 0 λ2N−1∗1 ψ∗1,1

......

......

......

......

......

λ 2Nψ1,N λNψ2,N λNψ3,N λ 4

Nψ1,N λ 3Nψ2,N λ 3

Nψ3,N · · · λ 2NN ψ1,N λ

2N−1N ψ2,N λ

2N−1N ψ3,N

−λ 2∗N ψ∗2,N λ ∗Nψ∗1,N 0 −λ 4∗

N ψ∗2,N λ 3∗N ψ∗1,N 0 · · · −λ 2N∗

N ψ∗2,N λ2N−1∗N ψ∗1,N 0

−λ 2∗N ψ∗3,N 0 λ ∗Nψ∗1,N −λ 4∗

N ψ∗3,N 0 λ 3∗N ψ∗1,N · · · −λ 2N∗

N ψ∗3,N 0 λ2N−1∗N ψ∗1,N

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣3N×3N

and G and F can be got by replacing the second andthird columns of H by Θ , respectively,

Θ = (−1)N+1 (ψ1,1,−ψ

∗2,1,−ψ

∗3,1, · · ·

ψ1,N ,−ψ∗2,N ,−ψ

∗3,N

)T.

(36)

4. Bright Vector Multi-Soliton Solutions

In this section, we will solve the Lax pair (2) and (3)with the zero seed potential and apply the N-times iter-ation of the Darboux transformation to yield the brightvector multi-soliton solution of (1).

4.1. Bright Vector One-Soliton Solution

With q1 = q2 = 0 as the seed solution and λ =λ1, we can get the following general solution of Laxpair (2) and (3):

Ψ1 =

ψ1,1ψ2,1ψ3,1

=

α1 eθ1

β1 e−θ12

δ1 e−θ12

, (37)

where the phase θ1 = i6γ2

(2µ−λ 2

1

)(2γx+λ 2

1 t−2µt);α1, β1, and δ1 are all arbitrary complex constants.

Substituting solution (37) into the N-times-iteratedpotential formula (34) and (35) for N = 1, we can ob-tain the bright one-soliton solution of (1):

q1[1] = 2√

2|β1||ξ1η1|eiϕ1 (38)

·[

γ

((|β1|2 + |δ1|2

){(ξ

21 +η

21

)· cosh

[32(θ1 +θ

∗1 )− ln

|β1|2 + |δ1|2

|α1|2

]+(ξ 2

1 −η21 )}) 1

2]−1

,

q2[1] = 2√

2|δ1||ξ1η1|eiζ1 (39)

·[

γ

((|β1|2 + |δ1|2

){(ξ

21 +η

21

)cosh

[32(θ1 +θ

∗1 )− ln

|β1|2 + |δ1|2

|α1|2

]+(ξ 2

1 −η21 )}) 1

2]−1

,

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

(a) (b)

Fig. 1 (colour online). Bright vector solitons via solutions (38) and (39). The parameters of relevant physical quantities areα1 = 2− i, β1 = 1, δ1 =−2, λ1 =−2.2−1.2i, γ = 2, and µ = 1.

where ξ1 and η1 are the real and imaginary part of λ1,respectively, and

ϕ1 =− i2

ln

{e

32 (θ1−θ∗1 )α1β

∗1

[λ1(β1β

∗1 +δ1δ

∗1 )

+λ∗1 α1α

∗1 e

32 (θ1+θ∗1 )

]3{

α∗1 β1

[λ∗1 (β1β

∗1

+δ1δ∗1 )+λ1α1α

∗1 e

32 (θ1+θ∗1 )

]3}−1

},

ζ1 =− i2

ln

{e

32 (θ1−θ∗1 )α1β

∗1

[λ1(β1δ

∗1 +δ1δ

∗1 )

+λ∗1 α1α

∗1 e

32 (θ1+θ∗1 )

]3{

α∗1 δ1

[λ∗1 (β1β

∗1

+δ1δ∗1 )+λ1α1α

∗1 e

32 (θ1+θ∗1 )

]3}−1

}.

From the above solutions (38) and (39), the brightvector solitons are characterized by four arbitrary com-plex parameters α1, β1, δ1, λ1, and two real parametersµ and γ . The amplitudes of bright solitons in the firstand second components are given as

A1 =2|η1β1|

γ√|β1|2 + |δ1|2

,

A2 =2|η1δ1|

γ√|β1|2 + |δ1|2

.

(40)

The velocity, initial phase, and width of bright vectorsolitons in two components are, respectively,

v =1γ

(η

21 −ξ

21 +2µ

),

ε =γ

ξ1η1ln|β1|2 + |δ1|2

|α1|2,

W =2ξ1η1

γ.

The bell profiles of vector bright solitons in two com-ponents are plotted in Figure 1.

4.2. Bright Vector Two-Soliton Solution

In order to obtain the bright vector two-soliton solu-tion, we adopt two sets of basic solutions of (2) and (3)with q1 = q2 = 0 for two different eigenvalues λ1 andλ2:

Ψk =

ψ1,kψ2,kψ3,k

=

αk eθk

βk e−θk2

δk e−θk2

, (k = 1,2) , (41)

where the phase θk = i6γ2

(2µ−λ 2

k

)[2γx +(λ 2

k −2µ)t]; αk, βk, and δk are all arbitrary complex con-stants. With the substitution of solutions (41) into (34)and (35) for N = 2, the bright vector two-soliton solu-tion can be derived as

q1[2] =−[

2iµγ

G2

H2−2

(G2

H2

)x

], (42)

q2[2] =−[

2iµγ

F2

H2−2

(F2

H2

)x

], (43)

where

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

H2 =

∣∣∣∣∣∣∣∣∣∣∣∣

λ 21 ψ1,1 λ1ψ2,1 λ1ψ3,1 λ 4

1 ψ1,1 λ 31 ψ2,1 λ 3

1 ψ3,1

−λ 2∗1 ψ∗2,1 λ ∗1 ψ∗1,1 0 −λ 4∗

1 ψ∗2,1 λ 3∗1 ψ∗1,1 0

−λ 2∗1 ψ∗3,1 0 λ ∗1 ψ∗1,1 −λ 4∗

1 ψ∗3,1 0 λ 3∗1 ψ∗1,1

λ 22 ψ1,2 λ2ψ2,2 λ2ψ3,2 λ 4

2 ψ1,2 λ 32 ψ2,2 λ 3

2 ψ3,2

−λ 2∗2 ψ∗2,2 λ ∗2 ψ∗1,2 0 −λ 4∗

2 ψ∗2,2 λ 3∗2 ψ∗1,2 0

−λ 2∗2 ψ∗3,2 0 λ ∗2 ψ∗1,2 −λ 4∗

2 ψ∗3,2 0 λ 3∗2 ψ∗1,2

∣∣∣∣∣∣∣∣∣∣∣∣6×6

G2 =

∣∣∣∣∣∣∣∣∣∣∣∣

λ 21 ψ1,1 −ψ1,1 λ1ψ3,1 λ 4

1 ψ1,1 λ 31 ψ2,1 λ 3

1 ψ3,1

−λ 2∗1 ψ∗2,1 ψ∗2,1 0 −λ 4∗

1 ψ∗2,1 λ 3∗1 ψ∗1,1 0

−λ 2∗1 ψ∗3,1 ψ∗3,1 λ ∗1 ψ∗1,1 −λ 4∗

1 ψ∗3,1 0 λ 3∗1 ψ∗1,1

λ 22 ψ1,2 −ψ1,2 λ2ψ3,2 λ 4

2 ψ1,2 λ 32 ψ2,2 λ 3

2 ψ3,2

−λ 2∗2 ψ∗2,2 ψ∗2,2 0 −λ 4∗

2 ψ∗2,2 λ 3∗2 ψ∗1,2 0

−λ 2∗2 ψ∗3,2 ψ∗3,2 λ ∗2 ψ∗1,2 −λ 4∗

2 ψ∗3,2 0 λ 3∗2 ψ∗1,2

∣∣∣∣∣∣∣∣∣∣∣∣6×6

F2 =

∣∣∣∣∣∣∣∣∣∣∣∣

λ 21 ψ1,1 λ1ψ2,1 −ψ1,1 λ 4

1 ψ1,1 λ 31 ψ2,1 λ 3

1 ψ3,1

−λ 2∗1 ψ∗2,1 λ ∗1 ψ∗1,1 ψ∗2,1 −λ 4∗

1 ψ∗2,1 λ 3∗1 ψ∗1,1 0

−λ 2∗1 ψ∗3,1 0 ψ∗3,1 −λ 4∗

1 ψ∗3,1 0 λ 3∗1 ψ∗1,1

λ 22 ψ1,2 λ2ψ2,2 −ψ1,2 λ 4

2 ψ1,2 λ 32 ψ2,2 λ 3

2 ψ3,2

−λ 2∗2 ψ∗2,2 λ ∗2 ψ∗1,2 ψ∗2,2 −λ 4∗

2 ψ∗2,2 λ 3∗2 ψ∗1,2 0

−λ 2∗2 ψ∗3,2 0 ψ∗3,2 −λ 4∗

2 ψ∗3,2 0 λ 3∗2 ψ∗1,2

∣∣∣∣∣∣∣∣∣∣∣∣6×6

.

Figure 2 displays the elastic collision of bright vec-tor solitons S1 and S2. In this figure, like the scalarsoliton, bright vector solitons S1 and S2 do not affecteach other only by a phase shift along with the con-served energy. They collide with each other and ex-hibit the particle properties. In contrast, Figure 3 showsthat the bright vector solitons S1 and S2 between twocomponents undergo the partial energy exchange ininelastic collision. In fact, the complete energy ex-change of two bright vector solitons also takes placebetween two components for suitable choice of param-eters. In addition, with the vanishing boundary condi-tions q j|x→±∞→ 0 ( j = 1,2), one can observe that thetotal energy of (1) is conserved from the integral ofmotion for (1), that is,∫ +∞

−∞

(|q1|2 + |q2|2

)dx = constant.

As illustrated in Figures 2 and 3, such elastic andinelastic collision properties of bright vector solitonscould be used in the construction of logic gates, the im-plementation of all-optical switching, and the design ofquantum information processors.

4.3. Bright Vector N-Soliton Solution

Based on the above procedure, we can successivelyimplement the Darboux transformation N times to gen-

erate the bright vector N-soliton solution of (1). Bysolving the linear equations (2) and (3) with the seedsolution q1 = q2 = 0 for N different spectral parame-ters λ = λk (1 6 k 6 N), we can take a set of linearlyindependent solutions

Ψk =

ψ1,kψ2,kψ3,k

=

αk eθk

βk e−θk2

δk e−θk2

,

(k = 1,2, . . . ,N) ,

(44)

where the phase θk = i6γ2 (2µ−λ 2

k )[2γx+(λ 2k −2µ)t];

αk, βk, and δk are all arbitrary complex constants.With the substitution solutions (44) into (34) and (35),the bright vector N-soliton solution of (1) can be ob-tained by virtue of the symbolic computation plat-form. Using the bright vector N-soliton solution,the interaction behaviours among and more collidingbright vector solitons can be studied such as com-plete or partial energy switching between two compo-nents.

Taking N = 3 for example, we adopt three differentspectral parameters λ1, λ2, and λ3 and correspondingthree linearly independent solutionsΨ1,Ψ2, andΨ3. Wecan study the collision behaviours among three brightvector solitons in two components. Figures 4 and 5 il-

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

(b)(a)

Fig. 2 (colour online). Elastic collision between two bright vector solitons. (a) Intensity profiles of the bright two-solitonsolutions (42) and (43). (b) Contour plots of the intensity versus t and x. The parameters of relevant physical quantities areα1 = 1− i, β1 = 1, δ1 = α2 = β2 = 1, δ2 =−1, λ1 = 1+ i, λ2 =−2+ i, γ = 1, and µ = 1.

lustrate the elastic and inelastic collisions among threebright vector solitons. In Figure 5, it can be easily ob-served that the intensity of bright vector soliton S1 getsenhanced and those of other two bright vector solitonsS2 and S3 get suppressed after the collision in the firstcomponent, while in the second component S1’ is sup-pressed and S2’ and S3’ are both enhanced after thecollision. Furthermore, one can check easily that theintensity of each bright vector soliton S j ( j = 1,2,3) isconserved in two components.

5. Conclusions

In this paper, we have investigated the coupledmixed derivative nonlinear Schrodinger equations (1),which describe the pulse propagation in the picosecondor femtosecond regime of birefringent optical fibers.A new type of Lax pair (2) and (3) associated with (1)has been derived from the Wadati–Konno–Ichikawa(WKI) spectral problem. In such a pair of linear equa-tions (2) and (3), the traces of U j and Vk are all

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

(a) (b)

Fig. 3 (colour online). Inelastic collision between two bright vector solitons. (a) Intensity profiles of the bright two-solitonsolutions (42) and (43). (b) Contour plots of the intensity versus t and x. The parameters of relevant physical quantities areα1 = 1− i, β1 = 1, δ1 =−2, α2 = β2 = δ2 = 1, λ1 = 1+ i, λ2 =−2+ i, γ = 1, and µ = 1.

zero, which would be useful for keeping Darboux co-variant reductions of (1). The Darboux transformationand iterative algorithm have been applied to this inte-grable coupled model, and the N-times iteration for-mulas (34) and (35) of the Darboux transformationhave been presented in terms of the determinant rep-resentation. With the zero potential as seed solutionand a given set of spectral parameters, the bright vec-tor N-soliton solution has been expressed as a com-

pact and transparent determinant by N linearly inde-pendent eigenfunctions of Lax pair (2) and (3). Wehave identified and discussed the interesting collisionproperties of bright vector solitons. It has been demon-strated that bright vector solitons can exhibit the stan-dard elastic and inelastic collisions. We have shownthat each vector soliton in two components can un-dergo the partial or complete energy switching in aninelastic collision.

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

(a) (b)

Fig. 4 (colour online). Elastic collision among three bright vector solitons. (a) Intensity profiles of the bright three-solitonsolutions (34) and (35) for N = 3. (b) Contour plots of the intensity versus t and x. The parameters of relevant physicalquantities are α1 = 1− i, β1 = 1, δ1 = α2 = β2 = 1, δ2 =−1, α3 = β3 = 1, δ3 =−1, λ1 = 1+ i, λ2 =−2+ i, λ3 =−3+ i,γ = 1, and µ = 1.

Based on the framework of the present paper, wehave the following miscellaneous remarks and futureworks.

(i) With the bilinear method, the bright N-soliton so-lution of (1) has been obtained in terms of the determi-nants and involves 3N complex parameters in [10]. Thedeterminants f and gi (i = 1,2) in the N-soliton solu-tion have compact expressions, where f is a 2N×2N

determinant and gi (i = 1,2) are both (2N + 1)×(2N + 1) determinants [10]. In the present paper, theN-soliton solution is expressed as a 3N×3N determi-nant by means of the Darboux transformation method.The determinant is generated by N complex eigenval-ues and N linearly independent eigenfunctions, and in-volves 4N complex parameters λk, αk, βk, and δk. Itseems that two types of the determinants are differ-

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

(a) (b)

Fig. 5 (colour online). Inelastic collision among three bright vector solitons. (a) Intensity profiles of the bright three-solitonsolutions (34) and (35) for N = 3. (b) Contour plots of the intensity versus t and x. The parameters of relevant physicalquantities are α1 = 1− i, β1 = 1, δ1 =−2, α2 = β2 = δ2 = 1, α3 = β3 = 1, δ3 =−1, λ1 = 1+ i, λ2 =−2+ i, λ3 =−3+ i,γ = 1, and µ = 1.

ent. The relation between them seems to be a nontrivialproblem and will be pursued in future works.

(ii) We have shown that the bright vector solitonsdisplay energy-exchange collision between two com-ponents in (1). Next, according to the determinantproperties of N-times Darboux iteration formula (3),we will investigate the asymptotic behaviour of thebright vector solitons for arbitrary N colliding vec-tor solitons. Such fascinating collision properties openpossibilities for future applications in the design of log-

ical gates, fiber directional couplers, and quantum in-formation processors.

(iii) Our focus will be put on the m-component cou-pled mixed derivative nonlinear Schrodinger equations

i q j t +q j xx + µ

(m

∑k=1

|qk|2)

q j

+ iγ

[(m

∑k=1

|qk|2)

q j

]x

= 0, ( j = 1,2, . . . ,m) .

(45)

H.-Q. Zhang · Darboux Transformation and N-soliton Solution for Coupled Modified NLSE

With the direct method, the bright N-soliton solutionof (45) has been obtained in the form of compact deter-minantal expressions [22]. Our Darboux transforma-tion method in the present paper can be easily gen-eralized to m-component (45), then the bright vec-tor N-soliton solution of (45) can be also derived interms of the determinant representation. Another in-teresting issue to be worth studying is the collisionprocess of arbitrary N colliding bright vector solitonswith complete or partial energy exchange among mcomponents.

(iv) In the present paper, we take zero as the seedsolution and obtain the bright N-soliton solution of (1)from the N-times iterative Darboux transformation.With nonvanishing background (e.g., monochromaticwave solution), the multi-periodic and breather solu-tions can be generated from the N-times-iterated po-tential formulas (34) and (35). Furthermore, we inferthat the multi-rogue wave solution of (1) can also be

derived with the N-times iterative Darboux transforma-tion under the nonzero background. In future works,our focus will be put on the periodic, breather, androgue wave solutions of (1) with nonvanishing back-ground.

Acknowledgement

This work has been supported by the Innova-tion Program of Shanghai Municipal Education Com-mission under Grant No. 12YZ105, by the Founda-tion of University Young Teachers Training Programof Shanghai Municipal Education Commission underGrant No. slg11029, by the Natural Science Founda-tion of Shanghai under Grant No. 12ZR1446800, Sci-ence and Technology Commission of Shanghai munic-ipality, by the National Natural Science Foundation ofChina under Grant Nos. 11201302 and 11171220.

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