Darboux Transformation and N-Soliton Solution for the Coupled Modified Nonlinear Schr¨odinger Equations

Darboux Transformation and N-Soliton Solution for the Coupled Modified Nonlinear Schr¨odinger Equations

Hai-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.67a,711 – 722 (2012) / DOI: 10.5560/ZNA.2012-0084

Received 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 is governed by the coupled mixed derivative nonlinear Schr¨odinger (CMDNLS) equations. A new type of Lax pair associated with such coupled equations is derived from the Wadati–Konno–Ichikawa spectral problem. The Darboux transformation method is applied to this integrable model, and the N-times iteration formula of the Darboux transformation is presented in terms of the compact de- terminant representation. Starting from the zero potential, the bright vectorN-soliton solution of CMDNLS equations is expressed as a compact determinant byNcomplex eigenvalues andNlin- early independent eigenfunctions. The collision mechanisms in two components shows that bright vector 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 fiber directional couplers, and quantum information processors.

Key words:Coupled Mixed Derivative Nonlinear Schr¨odinger Equations; Vector Soliton; Soliton Collision; Darboux Transformation.

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

1. Introduction

The coupled mixed derivative nonlinear Schr¨o- dinger equations in the dimensionless form

iq_{j t}+q_{j xx}+µ

2

∑

k=1

|q_k|²

! q_j

+iγ

"

2 k=1

∑

|q_k|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 qj =qj(x,t) (j =1,2) is the slowly varying complex envelope for polariza- tions, x and t appended to q_j denote partial differ- entiations, the parameters µ andγ are real constants as the nonlinearity and derivative nonlinearity coeffi- cients. For (1), the caseγ=0 is known as the coupled nonlinear Schr¨odinger (CNLS) equations [6,7], and µ=0 is the coupled derivative nonlinear Schr¨odinger

(CDNLS) equations governing the polarized Alfv´en waves 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 many local conservation laws [5,9], Lax pairs [2,5], and bilinear representations [3–5]. In addition, the exact bright N-soliton, dark and antidark soliton solutions have been presented by employing Hirota’s bilinear method [3–5,10].

In soliton theory, the Darboux transformation method has been a very effective tool to con- struct the exact analytical solutions of integrable NLEEs [11–16], especially the N-soliton solutions.

The purpose of the transformation is to produce new solutions 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 implemented on the symbolic computation system. With a succes- sive application of the Darboux transformation, theN-

© 2012 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

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

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

The structure of this paper will be arranged as fol- lows. In Section2, we will derive a new type of the Lax pair 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- tion3, we will present theN-times iterative potential formula of (1). In Section4, starting the zero seed so- lution, we will obtain the bright vectorN-soliton solu- tion of (1). Moreover, we will discuss energy-exchange collision behaviours between or among bright vector solitons in two components. Our conclusions will be given in Section5.

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 Darboux transformation. 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 associated with (1). In this subsection, based on the WKI spectral problem [12], we will present another new type of the Lax pair for (1).

Following the procedure generalizing the 2×2 WKI scheme to the 3×3 case, the Lax pair associated with (1) can be written as

Ψx=UΨ= λ²U₀+λU₁+U₂

Ψ, (2)

Ψt=VΨ= λ⁴V₀+λ³V₁+λ²V₂+λV₃+V₄ Ψ, (3) whereΨ= (ψ₁,ψ2,ψ3)^T(the superscript T denotes the vector transpose) is the vector eigenfunction,λ is the eigenvalue parameter,U_jandV_k(06j62; 06k64) are all 3×3 matrices to be determined. It is straighfor- ward matter to check that the integrability condition of (2) and (3),

U_t−V_x+ [U,V] =0, (4) is equivalent to (1) if the matricesUj andV_k(06j6 2; 06k64)take the form

U₀= 1 6γ





−2i 0 0

0 i 0

0 0 i



,

U₁=1 2





0 q1 q2

−q^∗₁ 0 0

−q^∗₂ 0 0



,

U2= µ 3γ





2i 0 0

0 −i 0

0 0 −i



,

(5)

V₀= 1 12γ²





−2i 0 0

0 i 0

0 0 i



,

V₁= 1 4γ





0 q1 q2

−q^∗₁ 0 0

−q^∗₂ 0 0



,

V4= µ² 3γ²





−2i 0 0

0 i 0

0 0 i



,

(6)

V₂= µ 3γ²





2i 0 0

0 −i 0

0 0 −i





+i 4





|q₁|²+|q₂|² 0 0 0 −|q₁|² −q^∗₁q2

0 −q₁q^∗₂ −|q₂|²



, (7)

V₃=γ 2





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

q^∗₂(|q₁|²+|q₂|²) 0

−q₂(|q₁|²+|q₂|²) 0

0





− µ 2γ





0 q₁ q₂

−q^∗₁ 0 0

−q^∗₂ 0 0



+i 2





0 q_1x q_2x q^∗_1x 0 0 q^∗_2x 0 0



, (8)

where the asterisk denotes the complex conjugate.

Remark. The above Lax pair (2) and (3) associated with (1) is different from those in [2,5]. Compared with the results in [2,5], the main difference is that the traces of matricesU_jandV_k(06j62; 06k64) in Lax pair (2) and (3) are all zero. On the basis of this matrix property, we will construct the Darboux trans- formation for (1).

2.2. Darboux Transformation

The Darboux transformation is a special gauge transformation which keeps the Lax pair invariant. In view of the feature that the space part (2) is quadratic about the spectral parameter λ, we take the Darboux transformation for (2) and (3) as the form

Ψb=DΨ=

2

∑

j=0

λ^2−j





a_1,j b_1,j b_2,j c1,j a2,j d1,j

c_2,j d_2,j a_3,j



Ψ, (9) which is in terms of the second-order polynomial ofλ, wherea_k,j,b_{`,j</sub>,c<sub>`,j}, andd<sub>`,j</sub>(k=1,2,3;`=1,2; 06

j62)are all functions ofxandtto be determined.

In order to keep the invariance of Lax pair (2) and (3) under the Darboux transformation (9), it re- quires thatΨb also satisfies the same linear eigenvalue problem (2) and (3) withU_j(06 j62)andV_k(06 k64), respectively, replaced byUb_jandVb_k. Thus, the original potentialsq1andq2are transformed into new potentials ˆq₁and ˆq₂, and the Darboux matrixDis re- quired to satisfy

D_x=U Db −DU, D_t=V Db −DV. (10) With use of the Darboux matrix (9), from (10) and by comparing the coefficients ofλ, we can directly com- pute

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

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

b_1,2=b_2,2=c_1,2=c_2,2=d_1,2=d_2,2=0, (12) a1,1=a2,1=d1,1=d2,1=d3,1=0, (13) a1,2=a2,2=a3,2=−1, (14)

ˆ

q1=q1+2iµ

γ b1,1−2b1,1x, ˆ

q₂=q₂+2iµ

γ b_2,1−2b_2,1x,

(15)

qˆ^∗₁=q^∗₁+2iµ

γ c1,1+2c_1,1x, ˆ

q^∗₂=q^∗₂+2iµ

γ c_2,1+2c_2,1x.

(16)

Thus, the Darboux matrixDcan be expressed as:

D=λ²





a_1,0 0 0 0 a2,0 d1,0

0 d_2,0 a_3,0





+λ





0 b_1,1 b_2,1 c_1,1 0 0 c2,1 0 0





+





−1 0 0

0 −1 0

0 0 −1



.

(17)

Next, our work is to determine the rest entriesak,0, b_{`,1</sub>,c<sub>`,1}, andd<sub>`,0</sub>(k=1,2,3;`=1,2)inD. From the knowledge about the kernel of the Darboux transfor- mation, withΨ1= (ψ₁,ψ2,ψ3)^T as a solution of Lax pair (2) and (3) forλ=λ₁, the undermined entries can be expressed in terms of the eigenvalue and eigenfunc- tion as

a1,0= 1

∆₁ λ₁^∗|ψ₁|²+λ₁|ψ₂|²+λ₁|ψ₃|²

, (18)

a2,0= 1

∆2

|λ₁|²|ψ₁|²+λ₁^2∗|ψ₂|²+λ₁²|ψ₃|²

, (19)

a_3,0= 1

∆2

|λ₁|²|ψ₁|²+λ₁²|ψ₂|²+λ₁^2∗|ψ₃|²

, (20)

d_1,0= 1

∆2

λ₁^2∗−λ₁² ψ2ψ₃^∗, d_2,0= 1

∆2

λ₁^2∗−λ₁² ψ3ψ₂^∗,

(21)

b_1,1= 1

∆1

λ₁^2∗−λ₁² ψ1ψ₂^∗, b_2,1= 1

∆1

λ₁^2∗−λ₁² ψ1ψ₃^∗,

(22)

c_1,1= 1

∆1

λ₁^2∗−λ₁² ψ2ψ₁^∗, c_2,1= 1

∆1

λ₁^2∗−λ₁² ψ3ψ₁^∗,

(23)

where

∆₁=|λ₁|² λ₁|ψ₁|²+λ₁^∗|ψ₂|²+λ₁^∗|ψ₃|² ,

∆2=|λ₁|² λ₁^2∗|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 the constraints (24).

To this state, we have finished the construction of the Darboux transformation for (1). Under the transfor- mation (17), the linear spectral problem (2) and (3) is invariant. At the same time, the new solution (qˆ₁,qˆ₂) can be obtained by choosing a special eigenvalue λ1

and corresponding eigenfunctions (ψ₁,ψ2,ψ3) from a seed solution. Therefore, we come to the following conclusion:

Assume thatΨ1= (ψ₁,ψ2,ψ3)^Tis a solution of Lax pair (2) and (3) withλ=λ1. The Darboux transforma- tion (Ψ,q₁,q₂)→(Ψb,qˆ₁,qˆ₂)of (1) is constructed by Transformation (17) with entries defined in (18) – (23).

The potential transformation between the new and original ones is given by

ˆ

q₁=q₁+2 λ₁^2∗−λ₁²

· iµ

γ ∆1

ψ1ψ₂^∗− 1

∆1

ψ1ψ₂^∗

x

, (25)

ˆ

q₂=q₂+2 λ₁^2∗−λ₁²

· iµ

γ ∆₁ψ1ψ₃^∗− 1

∆₁ψ1ψ₃^∗

x

. (26)

3.N-Times Iteration of the Darboux Transformation

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

polynomial inλ: D_n(x,t,λ) =

2N

∑

k=0

Γ_kλ^k (27)

=

N n=1

∑





a_1,2nλ²ⁿ b1,2n−1λ²ⁿ⁻¹ b2,2n−1λ²ⁿ⁻¹ c1,2n−1λ²ⁿ⁻¹ a2,2nλ²ⁿ d1,2nλ²ⁿ c2,2n−1λ²ⁿ⁻¹ d2,2nλ²ⁿ a3,2nλ²ⁿ





+ (−1)^NI,

whereIis the 3×3 identity matrix andΓ_k(06k62N) the coefficient matrix ofλ;Γ2n−1andΓ2n(06n6N) have the following structures:

Γ0= (−1)^NI, Γ2n−1=





0 b1,2n−1 b2,2n−1

c1,2n−1 0 0

c2,2n−1 0 0



,

Γ_2n=





a_1,2n 0 0 0 a2,2n d1,2n

0 d_2,2n a_3,2n



.

(28)

Let us take a set of linearly independent solutions Ψk= (ψ_1,k,ψ2,k,ψ_3,k)^Tof (2) and (3) with(q₁,q₂)for different spectral parametersλ=λ_k(16k6N), i. e., Ψk= (ψ_1,k,ψ2,k,ψ3,k)^T(16k6N)satisfy the linear equations (2) and (3):

[∂_x−U(λ =λ_k)]Ψ_k=0,

[∂_t−V(λ=λ_k)]Ψ_k=0. (29) We also introduce a set of orthogonal vectors

Φ_k⁽¹⁾= −ψ_2,k^∗ ,ψ_1,k^∗ ,0T

Φ_k⁽²⁾= −ψ_3,k^∗ ,0,ψ_1,k^∗ T

,

(30) which satisfy the orthogonality condition

D

Ψk|Φ_k^(`)E

=Ψ_k^†Φ_k<sup>(`)</sup>=0, (16k6N;`=1,2),

(31) where the dagger † signifies the Hermitian conjugate.

From the knowledge of the Darboux transformation, the following relations hold:

D_n(x,t,λ)|_λ=λ

kΨk=0, Dn(x,t,λ)|_λ=λ^∗

kΦ_k<sup>(`)</sup>=0, (16k6N;`=1,2),

(32)

which can be rewritten in a matrix form

(Γ₁,Γ₂,· · ·,Γ_2n)W_n=−B, (33) where

B=Γ0

Ψ1,Ψ₂,· · ·,ΨN,Φ₁⁽¹⁾,Φ₂⁽¹⁾,· · ·,Φ_N⁽¹⁾,Φ₁⁽²⁾,Φ₂⁽²⁾,· · ·,Φ_N⁽²⁾ ,

Wn=













λ1Ψ1 · · ·λNΨN λ₁^∗Φ⁽¹⁾₁ · · · λN^∗ΦN⁽¹⁾ λ₁^∗Φ₁⁽²⁾ · · · λN^∗ΦN⁽²⁾

λ₁²Ψ1· · ·λN²ΨN λ₁^2∗Φ₁⁽¹⁾· · ·λN^2∗ΦN⁽¹⁾ λ₁^2∗Φ₁⁽²⁾· · ·λN^2∗ΦN⁽²⁾

..

. · · · .

.. ..

. · · · .

.. ..

. · · · .

.. λ₁^2NΨ1· · ·λ_N^2NΨNλ₁^2N∗Φ₁⁽¹⁾· · ·λN^2N∗ΦN⁽¹⁾λ₁^2N∗Φ₁⁽²⁾· · ·λN^2N∗ΦN⁽²⁾











 .

Substituting (28) into linear algebraic (33), we can obtain the expressions of a_j,2n,d<sub>`,2n</sub>,b`,2n−1,c`,2n−1

(j=1,2,3;`=1,2; 16n6N)by use of Cramer’s

rule. Therefore, the potential formula ofN-times itera- tive Darboux transformation for (1) is given by q₁[N] =q₁+ (−1)^N+1

2iµ

γ b_1,1−2b_1,1x

=q₁+ (−1)^N+1 2iµ

γ G H−2

G H

x

,

(34)

q₂[N] =q₂+ (−1)^N+1 2iµ

γ b_2,1−2b_2,1x

=q2+ (−1)^N+1 2iµ

γ F H−2

F H

x

,

(35)

with

H=

λ₁²ψ1,1 λ1ψ2,1 λ1ψ3,1 λ₁⁴ψ1,1 λ₁³ψ2,1 λ₁³ψ3,1 · · · λ₁^2Nψ1,1 λ₁^2N−1ψ2,1 λ₁^2N−1ψ3,1

−λ₁^2∗ψ_2,1^∗ λ₁^∗ψ_1,1^∗ 0 −λ₁^4∗ψ_2,1^∗ λ₁^3∗ψ_1,1^∗ 0 · · · −λ₁^2N∗ψ_2,1^∗ λ₁^2N−1∗ψ_1,1^∗ 0

−λ₁^2∗ψ_3,1^∗ 0 λ₁^∗ψ_1,1^∗ −λ₁^4∗ψ_3,1^∗ 0 λ₁^3∗ψ_1,1^∗ · · · −λ₁^2N∗ψ_3,1^∗ 0 λ₁^2N−1∗ψ_1,1^∗

... ... ... ... ... ... ... ... ... ...

λ_N²ψ1,N λNψ2,NλNψ3,N λ_N⁴ψ1,N λ_N³ψ2,N λ_N³ψ3,N · · · λ_N^2Nψ1,N λ_N^2N−1ψ2,N λ_N^2N−1ψ3,N

−λ_N^2∗ψ_2,N^∗ λ_N^∗ψ_1,N^∗ 0 −λ_N^4∗ψ_2,N^∗ λ_N^3∗ψ_1,N^∗ 0 · · · −λ_N^2N∗ψ_2,N^∗ λ_N^2N−1∗ψ_1,N^∗ 0

−λ_N^2∗ψ_3,N^∗ 0 λ_N^∗ψ_1,N^∗ −λ_N^4∗ψ_3,N^∗ 0 λ_N^3∗ψ_1,N^∗ · · · −λ_N^2N∗ψ_3,N^∗ 0 λ_N^2N−1∗ψ_1,N^∗ 3N×3N

andGandF can be got by replacing the second and third columns ofHbyΘ, respectively,

Θ= (−1)^N+1 ψ_1,1,−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 theN-times iter- ation of the Darboux transformation to yield the bright vector 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 Lax pair (2) and (3):

Ψ₁=



 ψ_1,1 ψ2,1

ψ3,1



=







 α1e^θ1 β₁e⁻

θ1

2

δ1e⁻

θ₁ 2









, (37)

where the phaseθ₁= ⁱ

6γ² 2µ−λ₁²

(2γx+λ₁²t−2µt);

α₁,β₁, andδ₁are all arbitrary complex constants.

Substituting solution (37) into theN-times-iterated potential formula (34) and (35) forN=1, we can ob- tain the bright one-soliton solution of (1):

q₁[1] =2√

2|β₁||ξ₁η1|e^iϕ1 (38)

·

γ

|β₁|²+|δ₁|²

ξ₁²+η₁²

·cosh 3

2(θ₁+θ₁^∗)−ln|β₁|²+|δ₁|²

|α₁|²

+ (ξ₁²−η₁²)

¹₂−1

, q₂[1] =2√

2|δ₁||ξ₁η1|e^iζ1 (39)

·

γ

|β₁|²+|δ₁|²

ξ₁²+η₁²

cosh 3

2(θ₁+θ₁^∗)−ln|β₁|²+|δ₁|²

|α₁|²

+ (ξ₁²−η₁²)

¹₂−1

,

(a) (b)

Fig. 1 (colour online). Bright vector solitons via solutions (38) and (39). The parameters of relevant physical quantities are α1=2−i,β1=1,δ1=−2,λ1=−2.2−1.2i,γ=2, andµ=1.

whereξ1andη1are the real and imaginary part ofλ1, respectively, and

ϕ₁=−i 2ln

(

e³²(θ₁−θ₁^∗)α₁β₁^∗ h

λ₁(β₁β₁^∗+δ₁δ₁^∗) +λ₁^∗α₁α₁^∗e³²(θ₁+θ₁^∗)i3

α₁^∗β₁ h

λ₁^∗(β₁β₁^∗ +δ1δ₁^∗) +λ1α1α₁^∗e³²(θ1+θ₁^∗)i3−1)

,

ζ₁=−i 2ln

(

e³²(θ₁−θ₁^∗)α₁β₁^∗ h

λ₁(β₁δ₁^∗+δ₁δ₁^∗) +λ₁^∗α₁α₁^∗e³²(θ₁+θ₁^∗)i3

α₁^∗δ₁ h

λ₁^∗(β₁β₁^∗ +δ1δ₁^∗) +λ1α1α₁^∗e³²(θ1+θ₁^∗)i3−1)

.

From the above solutions (38) and (39), the bright vector solitons are characterized by four arbitrary com- plex parametersα1,β1,δ1,λ1, and two real parameters µandγ. The amplitudes of bright solitons in the first and second components are given as

A₁= 2|η₁β1| γp

|β₁|²+|δ₁|², A₂= 2|η₁δ₁|

γp

|β₁|²+|δ₁|².

(40)

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

v=1

γ η₁²−ξ₁²+2µ ,

ε= γ ξ1η1

ln|β₁|²+|δ₁|²

|α₁|² , W =2ξ₁η1

γ .

The bell profiles of vector bright solitons in two com- ponents are plotted in Figure1.

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) withq1=q2=0 for two different eigenvaluesλ1and λ2:

Ψ_k=



 ψ1,k

ψ_2,k ψ3,k



=







 αke^θk βke⁻

θ_k 2

δ_ke⁻

θ_k 2









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

where the phase θk = ⁱ

6γ² 2µ−λ_k²

[2γx +(λ_k²− 2µ)t]; αk, βk, and δk are all arbitrary complex con- stants. With the substitution of solutions (41) into (34) and (35) forN=2, the bright vector two-soliton solu- tion can be derived as

q₁[2] =− 2iµ

γ G2

H₂−2 G2

H₂

x

, (42)

q₂[2] =− 2iµ

γ F2

H₂−2 F2

H₂

x

, (43)

where

H₂=

λ₁²ψ1,1 λ1ψ2,1 λ1ψ3,1 λ₁⁴ψ1,1 λ₁³ψ2,1 λ₁³ψ3,1

−λ₁^2∗ψ_2,1^∗ λ₁^∗ψ_1,1^∗ 0 −λ₁^4∗ψ_2,1^∗ λ₁^3∗ψ_1,1^∗ 0

−λ₁^2∗ψ_3,1^∗ 0 λ₁^∗ψ_1,1^∗ −λ₁^4∗ψ_3,1^∗ 0 λ₁^3∗ψ_1,1^∗ λ₂²ψ1,2 λ2ψ2,2 λ2ψ3,2 λ₂⁴ψ1,2 λ₂³ψ2,2 λ₂³ψ3,2

−λ₂^2∗ψ_2,2^∗ λ₂^∗ψ_1,2^∗ 0 −λ₂^4∗ψ_2,2^∗ λ₂^3∗ψ_1,2^∗ 0

−λ₂^2∗ψ_3,2^∗ 0 λ₂^∗ψ_1,2^∗ −λ₂^4∗ψ_3,2^∗ 0 λ₂^3∗ψ_1,2^∗ 6×6

G2=

λ₁²ψ_1,1 −ψ_1,1 λ₁ψ_3,1 λ₁⁴ψ_1,1 λ₁³ψ_2,1 λ₁³ψ_3,1

−λ₁^2∗ψ_2,1^∗ ψ_2,1^∗ 0 −λ₁^4∗ψ_2,1^∗ λ₁^3∗ψ_1,1^∗ 0

−λ₁^2∗ψ_3,1^∗ ψ_3,1^∗ λ₁^∗ψ_1,1^∗ −λ₁^4∗ψ_3,1^∗ 0 λ₁^3∗ψ_1,1^∗ λ₂²ψ1,2 −ψ_1,2 λ2ψ3,2 λ₂⁴ψ1,2 λ₂³ψ2,2 λ₂³ψ3,2

−λ₂^2∗ψ_2,2^∗ ψ_2,2^∗ 0 −λ₂^4∗ψ_2,2^∗ λ₂^3∗ψ_1,2^∗ 0

−λ₂^2∗ψ_3,2^∗ ψ_3,2^∗ λ₂^∗ψ_1,2^∗ −λ₂^4∗ψ_3,2^∗ 0 λ₂^3∗ψ_1,2^∗ 6×6

F₂=

λ₁²ψ1,1 λ1ψ2,1 −ψ_1,1 λ₁⁴ψ1,1 λ₁³ψ2,1 λ₁³ψ3,1

−λ₁^2∗ψ_2,1^∗ λ₁^∗ψ_1,1^∗ ψ_2,1^∗ −λ₁^4∗ψ_2,1^∗ λ₁^3∗ψ_1,1^∗ 0

−λ₁^2∗ψ_3,1^∗ 0 ψ_3,1^∗ −λ₁^4∗ψ_3,1^∗ 0 λ₁^3∗ψ_1,1^∗ λ₂²ψ1,2 λ2ψ2,2 −ψ_1,2 λ₂⁴ψ1,2 λ₂³ψ2,2 λ₂³ψ3,2

−λ₂^2∗ψ_2,2^∗ λ₂^∗ψ_1,2^∗ ψ_2,2^∗ −λ₂^4∗ψ_2,2^∗ λ₂^3∗ψ_1,2^∗ 0

−λ₂^2∗ψ_3,2^∗ 0 ψ_3,2^∗ −λ₂^4∗ψ_3,2^∗ 0 λ₂^3∗ψ_1,2^∗ 6×6

.

Figure2displays the elastic collision of bright vec- tor solitons S₁ andS₂. In this figure, like the scalar soliton, bright vector solitonsS₁andS₂do not affect each 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, Figure3shows that the bright vector solitonsS₁ andS₂between two components undergo the partial energy exchange in inelastic collision. In fact, the complete energy ex- change of two bright vector solitons also takes place between two components for suitable choice of param- eters. In addition, with the vanishing boundary condi- tionsq_j|_x→±∞→0(j=1,2), one can observe that the total energy of (1) is conserved from the integral of motion for (1), that is,

Z +∞

−∞ |q₁|²+|q₂|²

dx=constant.

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

4.3. Bright Vector N-Soliton Solution

Based on the above procedure, we can successively implement the Darboux transformationNtimes to gen-

erate the bright vector N-soliton solution of (1). By solving the linear equations (2) and (3) with the seed solutionq₁=q₂=0 forN different spectral parame- tersλ=λk(16k6N), we can take a set of linearly independent solutions

Ψ_k=



 ψ_1,k ψ2,k

ψ3,k



=







 αke^θk β_ke⁻

θ_k 2

δ_ke⁻

θ_k 2







 ,

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

(44)

where the phaseθk= ⁱ

6γ²(2µ−λ_k²)[2γx+ (λ_k²−2µ)t];

αk, βk, and δk are all arbitrary complex constants.

With the substitution solutions (44) into (34) and (35), the bright vectorN-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 colliding bright vector solitons can be studied such as com- plete or partial energy switching between two compo- nents.

TakingN=3 for example, we adopt three different spectral parametersλ1,λ2, andλ3and corresponding three linearly independent solutionsΨ1,Ψ2, andΨ3. We can study the collision behaviours among three bright vector solitons in two components. Figures4and5il-

(b) (a)

Fig. 2 (colour online). Elastic collision between two bright vector solitons. (a) Intensity profiles of the bright two-soliton solutions (42) and (43). (b) Contour plots of the intensity versustandx. The parameters of relevant physical quantities are α1=1−i,β1=1,δ1=α2=β2=1,δ2=−1,λ1=1+i,λ2=−2+i,γ=1, andµ=1.

lustrate the elastic and inelastic collisions among three bright vector solitons. In Figure5, it can be easily ob- served that the intensity of bright vector solitonS₁gets enhanced and those of other two bright vector solitons S₂andS₃get suppressed after the collision in the first component, while in the second componentS1’ is sup- pressed and S₂’ and S₃’ are both enhanced after the collision. Furthermore, one can check easily that the intensity of each bright vector solitonS_j(j=1,2,3)is conserved in two components.

5. Conclusions

In this paper, we have investigated the coupled mixed derivative nonlinear Schr¨odinger equations (1), which describe the pulse propagation in the picosecond or 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 V_k are all

(a) (b)

Fig. 3 (colour online). Inelastic collision between two bright vector solitons. (a) Intensity profiles of the bright two-soliton solutions (42) and (43). (b) Contour plots of the intensity versustandx. The parameters of relevant physical quantities are α1=1−i,β1=1,δ1=−2,α2=β2=δ2=1,λ1=1+i,λ2=−2+i,γ=1, andµ=1.

zero, which would be useful for keeping Darboux co- variant reductions of (1). The Darboux transformation and iterative algorithm have been applied to this inte- grable coupled model, and the N-times iteration for- mulas (34) and (35) of the Darboux transformation have been presented in terms of the determinant rep- resentation. With the zero potential as seed solution and 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). We have identified and discussed the interesting collision properties of bright vector solitons. It has been demon- strated that bright vector solitons can exhibit the stan- dard elastic and inelastic collisions. We have shown that each vector soliton in two components can un- dergo the partial or complete energy switching in an inelastic collision.

(a) (b)

Fig. 4 (colour online). Elastic collision among three bright vector solitons. (a) Intensity profiles of the bright three-soliton solutions (34) and (35) forN=3. (b) Contour plots of the intensity versust andx. The parameters of relevant physical quantities areα1=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, andµ=1.

Based on the framework of the present paper, we have the following miscellaneous remarks and future works.

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

determinant and g_i(i = 1,2) are both (2N+1)×

(2N+1)determinants [10]. In the present paper, the N-soliton solution is expressed as a 3N×3Ndetermi- nant by means of the Darboux transformation method.

The determinant is generated byNcomplex eigenval- ues andNlinearly independent eigenfunctions, and in- volves 4N complex parametersλk, αk,βk, andδk. It seems that two types of the determinants are differ-

(a) (b)

Fig. 5 (colour online). Inelastic collision among three bright vector solitons. (a) Intensity profiles of the bright three-soliton solutions (34) and (35) forN=3. (b) Contour plots of the intensity versust andx. The parameters of relevant physical quantities areα1=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, andµ=1.

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

(ii) We have shown that the bright vector solitons display energy-exchange collision between two com- ponents in (1). Next, according to the determinant properties of N-times Darboux iteration formula (3), we will investigate the asymptotic behaviour of the bright vector solitons for arbitrary N colliding vec- tor solitons. Such fascinating collision properties open possibilities 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 them-component cou- pled mixed derivative nonlinear Schr¨odinger equations iq_{j t}+q_{j xx}+µ

m k=1

∑

|q_k|²

! q_j

+iγ

"

m k=1

∑

|q_k|²

! qj

#

x

=0, (j=1,2, . . . ,m). (45)

With the direct method, the bright N-soliton solution of (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 in terms of the determinant representation. Another in- teresting issue to be worth studying is the collision process of arbitraryN colliding bright vector solitons with complete or partial energy exchange among m components.

(iv) In the present paper, we take zero as the seed solution and obtain the brightN-soliton solution of (1) from the N-times iterative Darboux transformation.

With nonvanishing background (e.g., monochromatic wave solution), the multi-periodic and breather solu- tions can be generated from the N-times-iterated po- tential formulas (34) and (35). Furthermore, we infer that the multi-rogue wave solution of (1) can also be

derived with theN-times iterative Darboux transforma- tion under the nonzero background. In future works, our focus will be put on the periodic, breather, and rogue 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 Program of Shanghai Municipal Education Commission under Grant 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 of China under Grant Nos. 11201302 and 11171220.

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