Soliton Solutions, Pairwise Collisions and Partially Coherent Interactionsfor a Generalized (1+1)-Dimensional Coupled Nonlinear SchrödingerSystem via Symbolic Computation

Li-Li Lia, Bo Tiana,b,c, Hai-Qiang Zhanga, Xing Lüa, and Wen-Jun Liua

a School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122,Beijing 100876, China

b State Key Laboratory of Software Development Environment, Beijing University of Aeronauticsand Astronautics, Beijing 100083, China

c Key Laboratory of Optical Communication and Lightwave Technologies, Ministry of Education,Beijing University of Posts and Telecommunications, Beijing 100876, China

Reprint requests to B. T.; E-mail: gaoyt@public.bta.net.cn

Z. Naturforsch. 63a, 679 – 687 (2008); received April 15, 2008

With the aid of symbolic computation, we investigate a generalized (1+1)-dimensional couplednonlinear Schrödinger system with mixed nonlinear interactions, which has potential applications innonlinear optics and elastic solids. The exact analytical one-, two-, and three-soliton solutions arefirstly obtained by employing the bilinear method under two constraints. Some main propagation andinteraction properties of the solitons are discussed simultaneously. Moreover, some figures are plottedto graphically analyze the pairwise collisions and partially coherent interactions of three solitons.

Key words: Soliton Solution; Coupled Nonlinear Schrödinger System; Pairwise Collision;Partially Coherent Interaction; Symbolic Computation.

1. Introduction

Many nonlinear phenomena can be describedor modeled by nonlinear evolution equations(NLEEs) [1 – 4]. One typical NLEE is the non-linear Schrödinger (NLS) equation, a canonicalequation describing the evolution of the slowlyvarying amplitude of a wave propagating in weaklynonlinear and dispersive media [1]. However, if thereexist two or more waves in those media, the coupledNLS equations would be the relevant model [5]. Suchcoupled NLS models with physical interests havebeen widely studied in nonlinear wave motion inhydrodynamics, nonlinear optics, and even in solids[6, 7]. In [7], considering the propagation of highfrequency transverse waves in a weakly nonlinearand dispersive elastic solid medium, a generalized(1+1)-dimensional coupled NLS system is obtained as

iφτ +pφξ ξ +δ1|φ |2φ+δ2ψ2φ∗+δ3|ψ |2φ = 0,iψτ +pψξ ξ +δ1|ψ |2ψ+δ2φ2ψ∗+δ3|φ |2ψ = 0,

(1)

where φ(ξ ,τ) and ψ(ξ ,τ) are the complex amplitudes,the variables ξ and τ correspond to the normalizeddistance and time, respectively, the asterisk denotes

0932–0784 / 08 / 1000–0679 $ 06.00 c© 2008 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com

the complex conjugate, and δ1 �= 0, δ2, δ3 �= 0 and pare real constants describing the characteristics of themedium and of the interaction. The parameter p sig-nifies the relative sign of the group velocity disper-sion (GVD) [1], that is, a positive parameter p corre-sponds to the waves traveling in an anomalous GVDregion, and a negative p corresponds to waves in anormal GVD region (without loss of generality). Thecases, where the signs of the δ j ( j = 1,2,3) are not thesame, i. e., the so-called “mixed” interaction type [8],are more intriguing and may have important applica-tions in nonlinear phenomena.

Obviously, system (1) contains the following threecurrently important models arising from the opticalfiber system (where ξ and τ denote the transverse co-ordinate and the coordinate along the direction of prop-agation):

• The integrable coupled NLS equations of Man-akov type take the forms

iφτ + φξ ξ + δ (|φ |2 + |ψ |2)φ = 0,iψτ + ψξ ξ + δ (|ψ |2 + |φ |2)ψ = 0,

(2)

which govern the simultaneous propagations of two

680 L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System

orthogonal components of an electric field in opticalfibers [9] and are also important in describing the ef-fects of averaged random birefringence on an ortho-gonally polarized pulse in fibers [10]. For such a sys-tem, the exact soliton solutions, elastic and inelasticcollisions, together with the properties of solitons dur-ing collision including shape-changing intensity redis-tributions, amplitude-dependent phase shifts, and rela-tive separation distances have been stated in detail [11].Accordingly, in the present paper, we consider twoother cases.

• The system of coupled NLS equations in the formiφτ + φξ ξ + δ (|φ |2 + σ |ψ |2)φ = 0,iψτ + ψξ ξ + δ (|ψ |2 + σ |φ |2)ψ = 0,

(3)

describes the interactions of waves with different fre-quencies and orthogonally polarized components infibers [12], the propagations and nonlinear modula-tions of two electromagnetic waves with almost equalgroup velocities in birefringent fibers [13], and alsoplays an important role in the theory of soliton wave-length division multiplexing [14]. As presented in [15],system (3) does not pass the Painlevé test unless σ = 1.

• The integrable coupled NLS equations whichhave application for the propagations of pulses in anisotropic medium are written as [16, 17]

iφτ + φξ ξ + 2(|φ |2 + 2|ψ |2)φ −2ψ2φ∗ = 0,iψτ + ψξ ξ + 2(|ψ |2 + 2|φ |2)ψ −2φ2ψ∗ = 0,

(4)

the last terms of which are known as the coherent cou-pling terms and govern the energy exchange betweentwo axes of the fiber. For this model, the exact one- andtwo-soliton solutions have been constructed throughthe bilinear method [17]. In [18], the multi-soliton andthe soliton interaction behaviours have been presentedby virtue of the Darboux transformation.

In system (2), because the ratio between the para-meters of self-phase modulation (SPM) and cross-phase modulation (XPM) is considered to be equal,this type of integrable coupled NLS system is physi-cally valid only for the pulse propagation in a specialkind of optical fiber system [19]. But for wavelengthdivision multiplexing in single-model fibers and for thepropagations of linearly polarized pulses in birefrin-gent fibers, the ratio between SPM and XPM in theintegrable system, e. g., system (4), has to be 1/2 [14].In general, the coupled NLS system (1) with arbitrarycoefficients is not integrable.

As reported in [20], the partially coherent multi-soliton complexes [21, 22] propagating in photorefrac-tive media can exhibit high nonlinearity with extremelylow optical power. Since the multi-soliton complexesare special cases of the higher-order bright soliton so-lutions [11], the study of varying profiles of theseexperimentally observed multi-soliton complexes andtheir interesting collision behaviours will be facilitatedby searching for the higher-order soliton solutions ofNLS systems. Therefore, the investigation of coupledNLS equations both integrable and nonintegrable isnecessary and of considerable significance for thesetopics [11].

In the present paper, with the aid of symbolic com-putation [2 – 4], we focus on constructing solutions forsystem (1) by employing the bilinear method [23],where the procedure can be generalized in principleto higher-order soliton solutions. The exact analyticalone-, two-, and three-soliton solutions (including thesingular and bright soliton ones) are obtained undertwo constraints, I: δ2 = 0 and δ1 �= ±δ3; II: δ2 �= 0 andδ1 ±δ2 −δ3 = 0, recovering the aforementioned cases,especially when the ratio between the parameters ofSPM and XPM is not 1/2. Based on the obtained solu-tions, some main propagation and interaction proper-ties of the solitons will be discussed. In particular, wepoint out that the pairwise collision [11] and partiallycoherent interaction [20 – 22] features also persist insystem (1) as in the Manakov case.

The outline of the present paper is as follows: inSection 2, the exact analytical soliton solutions for sys-tem (1) will be given; in Section 3, propagation and in-teraction properties of the solitons will be investigatedthrough graphical illustration; Section 4 will serve fora summary.

2. Soliton Solutions for System (1)

By introducing the dependent variable transforma-tions

φ =GF

, ψ =HF

, (5)

where G(ξ ,τ) and H(ξ ,τ) are both complex differen-tiable functions and F(ξ ,τ) is a real one, system (1)can be decoupled into the bilinear forms

(iDτ + pD2ξ )(G ·F) = 0,

(iDτ + pD2ξ )(H ·F) = 0,

L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System 681(δ1|G|2 + δ3|H|2)G+ δ2H2G∗−pGD2ξ (F ·F) = 0,(δ1|H|2 + δ3|G|2)H + δ2G2H∗−pHD2ξ (F ·F) = 0.

(6)

Here Dmξ Dnτ is the bilinear operator [23] defined by

Dmξ Dnτ(a ·b)≡

(∂

∂ξ− ∂

∂ξ ′

)m ( ∂∂τ

− ∂∂τ ′

)n·a(ξ ,τ)b(ξ ′,τ ′)

∣∣∣ξ ′=ξ ,τ ′=τ

.

In the following, we will construct the exact analyticalone-, two-, and three-soliton solutions for system (1)by solving (6).

2.1. One-Soliton Solutions

To obtain the one-soliton solutions, we expand thefunctions G, H, and F as

G = εg1, H = εh1, F = 1 + ε2 f2, (7)

where ε is a formal expansion parameter that can betaken as 1 [23]. With the assumption that g1 and h1have solutions of the form

g1 = aeθ , h1 = beθ , θ = kξ + ωτ, (8)

where a, b, k, ω are arbitrary nonzero complex param-eters, we substitute expressions (7) into (6) and thenequate to zero the coefficients of the terms with thesame power of ε . The resulting set of equations canbe solved recursively to obtain the dispersive relationand function f2. Via symbolic computation, consider-ing the values of parameters δ1, δ2 and δ3, we obtaintwo sets of one-soliton solutions for system (1).

I-1: δ2 = 0 and δ1 �= ±δ3. Then

ω = ik2 p, f2 =|a|2(δ1 + δ3)2p(k1 + k∗1)

eθ+θ∗, |a|= |b|.

When p(δ1 + δ3) > 0, the one-soliton solution forsystem (1) written in conventional form is

φ =aRe(k)|a|

√2p

δ1 + δ3ei{pτ[Re(k)2−Im(k)2]+Im(k)ξ}

· sech[

Re(k)ξ −2pRe(k)Im(k)τ

+12

log|a|2(δ1 + δ3)

8pRe(k)2

],

ψ =ba

φ ;

(9)

when p(δ1 + δ3) < 0, then it is

φ =−aRe(k)

|a|

√ −2pδ1 + δ3

ei{pτ[Re(k)2−Im(k)2]+Im(k)ξ}

· csch[

Re(k)ξ −2pRe(k)Im(k)τ

+12

log−|a|2(δ1 + δ3)

8pRe(k)2

],

ψ =ba

φ .

(10)

II-1: δ2 �= 0 and δ1 ± δ2 − δ3 = 0. Then

ω = ik2 p, f2 =|a|2δ1(1 + |γ|2)

2p(k1 + k∗1)eθ+θ

∗, b = γa,

where γ is an arbitrary constant satisfying γγ∗ =δ1−δ3

δ2.

Consequently, the one-soliton solution for system (1)written in conventional form, when pδ1 > 0, is

φ =aRe(k)|a|

√2p

δ1(1 + |γ|2)ei{pτ[Re(k)2−Im(k)2]+Im(k)ξ}

· sech[

Re(k)ξ −2pRe(k)Im(k)τ

+12

log|a|2δ1(1 + |γ|2)

8pRe(k)2

],

ψ = γφ ;

(11)

when pδ1 < 0, then it is

φ =−aRe(k)

|a|

√−2p

δ1(1+|γ|2)ei{pτ[Re(k)2−Im(k)2]+Im(k)ξ}

· csch[

Re(k)ξ −2pRe(k)Im(k)τ

+12

log−|a|2δ1(1 + |γ|2)

8pRe(k)2

],

ψ = γφ .

(12)

It is obvious that solutions (9) and (11) are the brightsolitons, while solutions (10) and (12) are singular onthe lines:

I-1 : Re(k)ξ −2pRe(k)Im(k)τ

+12

log−|a|2(δ1 + δ3)

8pRe(k)2= 0,

II-1 : Re(k)ξ −2pRe(k)Im(k)τ

+12

log−|a|2δ1(1 + |γ|2)

8pRe(k)2= 0.

682 L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System(a)

-30

3Ξ-4

-20

24

Τ

0

1

2

�Φ�

-30

3Ξ

(b)

-30

3Ξ-4

-20

24

Τ

0

1

2

�Φ�

-30

3Ξ

Fig. 1. Intensity plots of the bright one-soliton solution (11). The choice of the parameters is: (a) a = 1, k = 1 + 0.3i; (b)a = 2, k = 2− i; the common parameters are γ = 2 and p = δ1 = 1.

From expressions (9) and (11) we know that the am-plitudes of the bright one-soliton in the first and secondcomponents are given by

I-1 : Aφ = Aψ =

∣∣∣∣∣Re(k)√

2p(δ1 + δ3)

∣∣∣∣∣ ,II-1 : Aφ =

∣∣∣∣∣Re(k)√

2pδ1(1 + |γ|2)

∣∣∣∣∣ ,Aψ =

∣∣∣∣∣γRe(k)√

2pδ1(1 + |γ|2)

∣∣∣∣∣ ,where the symbol A denotes the amplitude. The solitonvelocities in each component are equal which can beexpressed by 2 p Im(k). The positions of the solitonsare

I-1 :1

2Re(k)log

|a|2(δ1 + δ3)8pRe(k)2

,

II-1 :1

2Re(k)log

|a|2δ1(1 + |γ|2)8pRe(k)2

.

(13)

Based on the expressions above we can concludethat parameters p, δ1, δ3, γ , and the real part of k are to-gether responsible for the soliton amplitude; p and theimaginary part of k dominate the soliton velocity; p,δ1, δ3, γ , modulus of a and the real part of k determinethe position of soliton together. As depicted in Fig. 1,the soliton amplitude in Fig. 1b is higher than that inFig. 1a with the increase of Re(k). The soliton velocitychanges according to the value of Im(k).

2.2. Two-Soliton Solutions

For generating the two-soliton solutions, the func-tions G, H and F can be expanded as

G = εg1 + ε3g3,H = εh1 + ε3h3,F = 1 + ε2 f2 + ε4 f4.

(14)

With the assumptions of the solutions

g1 = a1eθ1 + a2eθ2 ,

h1 = b1eθ1 + b2eθ2 ,

g3 = a3eθ1+θ2+θ∗1 + a4eθ1+θ2+θ

∗2 ,

h3 = b3eθ1+θ2+θ∗1 + b4eθ1+θ2+θ

∗2 ,

θ j = k jξ + ω jτ ( j = 1,2),

(15)

and manipulation as aforementioned, the dispersive re-lation and functions f2 and f4 are found to be

ω j = ik2j p ( j = 1,2),f2 = M11eθ1+θ

∗1 + M12eθ1+θ

∗2

+ M21eθ2+θ∗1 + M22eθ2+θ

∗2 ,

f4 = M4eθ1+θ∗1 +θ2+θ

∗2 ,

where

a3 =(k1 − k2)(a1κ21 −a2κ11)

(k1 + k∗1)(k2 + k∗1)

,

a4 =(k2 − k1)(a2κ12 −a1κ22)

(k1 + k∗2)(k2 + k∗2)

,

Mjl =κ jl

(k j + k∗l )( j, l = 1,2),

M4 =|k1 − k2|2(κ11κ22 −κ12κ21)

(k1 + k∗1)(k2 + k∗2)|(k1 + k∗2)|2

.

(16)

According to the values of δ1, δ2, and δ3, various pa-rameters in expressions (16) are defined as follows:

L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System 683

I-2: δ2 = 0 and δ1 �= ±δ3. Then

κ jl =a ja∗l (δ1 + δ3)2p(k j + k∗l )

,

bm = γam (m = 1,2,3,4),|γ| = 1.

II-2: δ2 �= 0 and δ1 ± δ2 − δ3 = 0. Then

κ jl =a ja∗l δ1(1 + |γ|2)

2p(k j + k∗l ),

bm = γam (m = 1,2,3,4),γγ∗

=δ1 − δ3

δ2.

In the above cases, a j, k j ( j, l = 1,2), and γ are allarbitrary nonzero complex constants.

Therefore, through the transformations (5), the two-soliton solution for system (1) can be written as

φ =(

a1eθ1 + a2eθ2 + a3eθ1+θ2+θ∗1 + a4eθ1+θ2+θ

∗2

)/(

1 + M11eθ1+θ∗1 + M12eθ1+θ

∗2 + M21eθ2+θ

∗1

+ M22eθ2+θ∗2 + M4eθ1+θ

∗1 +θ2+θ

∗2

),

ψ = γφ .

(17)

2.3. Three-Soliton Solutions

It is possible to derive three-soliton solutions withthe general expressions for G, H and F as

G = εg1 + ε3g3 + ε5g5,H = εh1 + ε3h3 + ε5h5,F = 1 + ε2 f2 + ε4 f4 + ε6 f6,

(18)

where

g1 = a1eθ1 + a2eθ2 + a3eθ3 ,

h1 = b1eθ1 + b2eθ2 + b3eθ3 ,

g3 = a121eθ1+θ2+θ∗1 + a122eθ1+θ2+θ

∗2 + a233eθ2+θ3+θ

∗3

+ a232eθ2+θ3+θ∗2 + a133eθ1+θ3+θ

∗3 + a131eθ1+θ3+θ

∗1

+ a123eθ1+θ2+θ∗3 +a231eθ2+θ3+θ

∗1 +a132eθ1+θ3+θ

∗2 ,

h3 = b121eθ1+θ2+θ∗1 + b122eθ1+θ2+θ

∗2 + b233eθ2+θ3+θ

∗3

+ b232eθ2+θ3+θ∗2 + b133eθ1+θ3+θ

∗3 + b131eθ1+θ3+θ

∗1

+ b123eθ1+θ2+θ∗3 +b231eθ2+θ3+θ

∗1 +b132eθ1+θ3+θ

∗2 ,

g5 = a12eθ1+θ2+θ3+θ∗1 +θ

∗2 + a23eθ1+θ2+θ3+θ

∗2 +θ

∗3

+ a13eθ1+θ2+θ3+θ∗1 +θ

∗3 ,

h5 = b12eθ1+θ2+θ3+θ∗1 +θ

∗2 + b23eθ1+θ2+θ3+θ

∗2 +θ

∗3

+ b13eθ1+θ2+θ3+θ∗1 +θ

∗3 ,

f2 = M11eθ1+θ∗1 + M12eθ1+θ

∗2 + M13eθ1+θ

∗3

+ M21eθ2+θ∗1 + M22eθ2+θ

∗2 + M23eθ2+θ

∗3

+ M31eθ3+θ∗1 + M32eθ3+θ

∗2 + M33eθ3+θ

∗3 ,

f4 = M1212eθ1+θ2+θ∗1 +θ

∗2 + M1313eθ1+θ3+θ

∗1 +θ

∗3

+ M2323eθ2+θ3+θ∗2 +θ

∗3 + M1312eθ1+θ3+θ

∗1 +θ

∗2

+ M1213eθ1+θ2+θ∗1 +θ

∗3 + M2312eθ2+θ3+θ

∗1 +θ

∗2

+ M1223eθ1+θ2+θ∗2 +θ

∗3 + M1323eθ1+θ3+θ

∗2 +θ

∗3

+ M2313eθ2+θ3+θ∗1 +θ

∗3 ,

f6 = M6eθ1+θ∗1 +θ2+θ

∗2 +θ3+θ

∗3 ,

θ j = k jξ + ik2j pτ ( j = 1,2,3). (19)

By solving the resulting linear partial differential equa-tions recursively via symbolic computation, the param-eters in expressions (19) are determined to be

am jl =(km − k j)(am κ jl −a j κml)

(km + k∗l )(k j + k∗l )

,

bm jl =(km − k j)(bm κ jl −b j κml)

(km + k∗l )(k j + k∗l )

,

a jl =[(k1 − k2)(k2 − k3)(k3 − k1)(k∗j − k∗l )

][(k1 + k∗j)

·(k1 + k∗l )(k2 + k∗j )(k2 + k∗l )(k3 + k∗j)(k3 + k∗l )]−1

·[a3(κ1lκ2 j −κ1 jκ2l)−a2 (κ1lκ3 j −κ1 jκ3l)

+a1(κ2lκ3 j −κ2 jκ3l)],

b jl =[(k1 − k2)(k2 − k3)(k3 − k1)(k∗j − k∗l )

][(k1 + k∗j)

·(k1 + k∗l )(k2 + k∗j )(k2 + k∗l )(k3 + k∗j)(k3 + k∗l )]−1

·[b3(κ1lκ2 j −κ1 jκ2l)−b2 (κ1lκ3 j −κ1 jκ3l)

+b1(κ2lκ3 j −κ2 jκ3l)],

Mjl =κ jl

(k j + k∗l ),

Mm jlq =(km − k j)(k∗l − k∗q)(κmlκ jq −κmqκ jl )(km + k∗l )(km + k∗q)(k j + k

∗l )(k j + k

∗q)

,

684 L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System

M6 =(|k1 − k2|2|k1 − k3|2|k2 − k3|2

)/[(k1 + k∗1)(k2 + k

∗2)(k3 + k

∗3)|k1 + k∗2|2|k1

+ k∗3|2|k2 + k∗3|2]

· (κ12κ23κ31 −κ13κ22κ31 + κ13κ21κ32−κ11κ23κ32 −κ12κ21κ33 + κ11κ22κ33),

where m, j, l,q = 1,2,3, a j and k j are arbitrary nonzerocomplex constants. For different cases of δ1, δ2 andδ3, the various parameters given in above expressionsare defined as those in I-2 and II-2 for the two-solitonsolutions.

Therefore, the three-soliton solution for system (1)can be derived as

φ =g1 + g3 + g5

1 + f2 + f4 + f6,

ψ = γφ .(20)

The above procedure can be straightforwardly gen-eralized to construct the four-soliton and higher-ordersoliton solutions. Although the expressions will bequite lengthy with the increase of order, it can be ex-plicitly represented in terms of exponential functions(the details are omitted here).

3. Pairwise Collisions and Partially CoherentInteractions of the Bright Solitons

According to the solutions obtained in the previoussection, we will detailedly investigate the propagationand interaction characteristics of the bright solitons.

With the aid of symbolic calculations, by virtue ofsystem (1) and its complex conjugate equations, it iseasy to obtain the energy conservation law as

i∂

∂τ(|φ |2 + |ψ |2) =

∂∂ξ

[p(

φφ∗ξ −φ∗φξ + ψψ∗ξ −ψ∗ψξ)]

.

For the obtained two- and three-soliton solutions inSection 2, because the ratios of parameters a j to b j( j = 1,2,3) are equal, the collisions of the solitonsare elastic, and there is no energy exchange betweenthe two components except for hardly visible phaseshifts, whereas the interesting pairwise collision andpartially coherent interaction features persist in system(1) for three- or higher-order solitons as in the Man-akov case [11]. In the following, utilizing the three-soliton solution of case II, some figures will be plotted

to explicitly illustrate the pairwise collisions and thepartially coherent interactions.

Figure 2 displays the pairwise elastic collisions ofthree solitons with different amplitudes and veloci-ties expressed via solution (20). As analyzed above,the three solitons pass through each other unaffect-edly with only small phase shifts during the processof collision. Similar to the situation studied in [11], thecollision is indeed a pairwise one which can be seendirectly and clearly in Figure 2. We can learn fromFig. 2a that the centres of the three solitons collidewith each other simultaneously at the same spot withthe proper choice of parameters to ensure the propa-gation lines of the three solitons intersect at the samepoint. When we set Im(k1) = Im(k2) �= Im(k3), the cor-responding solitons S1 and S2 can propagate in parallel,while S3 does not. As demonstrated in Fig. 2b, S3, af-ter the first collision with S2, collides with S1. The lastplot in Fig. 2 gives the perfect illustration of the pair-wise collision among three solitons at three differentpositions. The reason why the three collisions occurin different forms can be deduced from the differentchoices of a j and k j ( j = 1,2,3), which affect the ve-locities and positions of the solitons significantly. Sim-ilar to the one-soliton case, if parameters p, δ1, and γare given, the velocities of the colliding solitons ex-pressed by solution (20) are determined by the imag-inary parts of k j, and the positions are related to themodulus of a j and the real parts of k j ( j = 1,2,3).

When the relative separation distance of the soli-tons is small enough, a partially coherent interactionwill take place, which is shown in Figure 3. Under thepartially coherent interaction, the three parallel soli-tons superpose nonlinearly [22], thus the three-solitoncomplex [11, 20] is formed. Figure 3 evidently showsthat the profile of the complex varies periodically, andthis strongly depends on the values of a j and k j ( j =1,2,3), which determine how the solitons spread out.As in the case of the one soliton, the modulus of a j andthe real parts of k j ( j = 1,2,3) influence the soliton po-sitions (see expression (13)), and further determine therelative distances of solitons. The smaller the relativedistance of solitons is, the more intense the mutual in-teraction will be. Consequently, the three-soliton com-plex will possess a rich variety of structures accord-ing to the varied propagation parameters of the single-soliton components.

In fact, a partially coherent interaction of the soli-tons exists not only when the solitons transmit in paral-lel but also when they collide. The difference between

L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System 685(a)

-20 0 20

Ξ

0

10

20

Τ

�Φ�

S1

S2

S3(b)

-20 0 20

Ξ

0

10

20

Τ

�Φ�

S1

S2

S3

(c)

-20 0 20

Ξ

0

10

20

Τ

�Φ�

S1

S2

S3

Fig. 2. Planform of pairwise elastic collisions of three soli-tons expressed by solution (20). The choice of the para-meters is: (a) a1 = 0.5, a2 = 1, k2 = 1 + i, k3 = 2 + 0.5i;(b) a1 = 1000, a2 = 0.1, k2 = 1+2i, k3 = 2; (c) a1 = 1000,a2 = 0.1, k2 = 1 + i, k3 = 2− 0.5i; the common parametersare k1 = 0.5+2i, γ =

√3, δ1 = 2, p = 1, and a3 = 2.

the two cases is that the interaction merely occurs dur-ing the collision process when the relative separationdistance of the solitons is small enough. As an exampleof this, Fig. 4 demonstrates the collision scenario be-tween a single soliton and a two-soliton complex. Dif-ferent from the case in Fig. 2b, the two parallel solitonsmerge together to form a two-soliton complex. Dur-ing the collision process, as exhibited in Fig. 4b, thepartially coherent interaction occurs among the threesolitons, then a three-soliton complex is formed. Theshape of the complex is variable and disappears aftercollision with the increase of relative separation dis-tance. It is worth mentioning that such an interactionalso exists in the collision process in Figure 2.

4. Summary

A generalized (1+1)-dimensional coupled nonlinearSchrödinger system with mixed nonlinear interactionshas been investigated, which has potential applica-tions in nonlinear optics and elastic solids. It is knownthat constructing the exact analytic three-soliton solu-tions is a very laborious and tedious task for NLS-typeequations. However, computerized symbolic computa-tion [2 – 4], a branch of artificial intelligence, is ableto improve drastically the computation ability to dealwith this problem exactly and algorithmically. Thuswith the help of symbolic computation, the one-, two-and three-soliton solutions for system (1) have been

686 L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System(a)

-5

0

5Ξ0

2

4

6

8

Τ0

1

2

�Φ�

-5

0

5Ξ

(b)

-5

0

5Ξ0

2

4

6

8

Τ0

1

2

�Φ�

-5

0

5Ξ

Fig. 3. Partially coherent interaction of three parallel solitons expressed by solution (20). The choice of the parameters is:(a) a1 = a3 = k3 = 1; (b) a1 = 8, a3 = 3, k3 = 3; the common parameters are a2 = 2, k1 = 1, k2 = 2, γ =

√3, δ1 = 2, and

p = 1.

(a)

-40 -20 20 40 Ξ

0.5

1

�Φ�(b)

-40 -20 20 40 Ξ

0.5

1

�Φ�

(c)

-40 -20 20 40 Ξ

0.5

1

�Φ�

Fig. 4. Intensity profiles showing the collision scenario be-tween a single soliton and a two-soliton complex describedby solution (20), at (a) τ = −12; (b) τ = 0; (c) τ = 12.The choice of the other parameters is: a1 = a3 = 1, a2 = 2,k1 = 1+ i, k2 = 2+ i, k3 = 1− i, γ =

√3, δ1 = 2, and p = 1.

derived through the bilinear method under two con-straints, and this procedure can be straightforwardlygeneralized to construct higher-order soliton solutions.It has been given in [17], that system (1) can pass thePainlevé test when the parameters satisfy: (i) δ1 = δ3,δ2 = 0 or (ii) δ3 = 2δ1, δ2 = ±δ1. System (1) undercondition (i) is the celebrated Manakov type that hasbeen investigated in many papers [9 – 11], while withcondition (ii), e. g., system (3), it is only a special sit-uation of case II considered in this paper, and the so-

lutions given in [17] are consistent with ours. Further-more, on the basis of the three-soliton solutions, thepairwise collisions and partially coherent interactionsof the solitons have been analyzed through graphicalillustration.

Acknowledgements

We would like to thank the referee, Prof. Y. T. Gao,Mr. T. Xu and Ms. J. Li for their valuable comments.

L.-L. Li et al. · (1+1)-Dimensional Coupled Nonlinear Schrödinger System 687

This work has been supported by the National Nat-ural Science Foundation of China under Grant Nos.60772023 and 60372095, by the Key Project of Chi-nese Ministry of Education (No. 106033), by the OpenFund of the State Key Laboratory of Software Devel-opment Environment, Beijing University of Aeronau-

tics and Astronautics under Grant No. SKLSDE-07-001, by the National Basic Research Program of China(973 Program) under Grant No. 2005CB321901, andby the Specialized Research Fund for the Doctoral Pro-gram of Higher Education, Chinese Ministry of Educa-tion (No. 20060006024).

[1] A. Hasegawa, Optical Solitons in Fibers, Springer,Berlin 1989; K. Porsezian and V. C. Kuriakose, Opti-cal Solitons: Theoretical and Experimental Challenges,Springer, Berlin 2003.

[2] M. P. Barnett, J. F. Capitani, J. Von Zur Gathen, andJ. Gerhard, Int. J. Quant. Chem. 100, 80 (2004); B. Tianand Y. T. Gao, Phys. Plasmas 12, 054701 (2005); Phys.Lett. A 340, 243 (2005); 340, 449 (2005); 362, 283(2007).

[3] G. Das and J. Sarma, Phys. Plasmas 6, 4394 (1999);Z. Y. Yan and H. Q. Zhang, J. Phys. A 34, 1785 (2001);Y. T. Gao and B. Tian, Phys. Plasmas 13, 112901(2006); Phys. Plasmas (Lett.) 13, 120703 (2006); Phys.Lett. A 349, 314 (2006); B. Tian, W. R. Shan, C. Y.Zhang, G. M. Wei, and Y. T. Gao, Eur. Phys. J. B (RapidNot.) 47, 329 (2005); B. Tian, G. M. Wei, C. Y. Zhang,W. R. Shan, and Y. T. Gao, Phys. Lett. A 356, 8 (2006).

[4] W. P. Hong, Phys. Lett. A 361, 520 (2007); B. Tian andY. T. Gao, Phys. Plasmas (Lett.) 12, 070703 (2005);Eur. Phys. J. D 33, 59 (2005); Y. T. Gao and B. Tian,Phys. Lett. A 361, 523 (2007); Europhys. Lett. 77,15001 (2007); L. L. Li, B. Tian, C. Y. Zhang, and T. Xu,Phys. Scr. 76, 411 (2007).

[5] H. A. Erbay, S. Erbay, and S. Dost, Int. J. Eng. Sci. 29,859 (1991); Y. Kajinaga, T. Tsuchida, and M. Wadati, J.Phys. Soc. Jpn. 67, 1565 (1998); S. C. Tsang and K. W.Chow, Math. Comput. Simul. 66, 551 (2004).

[6] J. J. Rasmussen and K. Rypdal, Phys. Scr. 33, 481(1986); I. Bakirtas and H. Demiray, Appl. Math. Com-put. 154, 747 (2004); B. Tian and Y. T. Gao, Phys. Lett.A 342, 228 (2005); 359, 241 (2006); B. Tian, Y. T.Gao, and H. W. Zhu, Phys. Lett. A 366, 223 (2007);X. Lü, H. W. Zhu, X. H. Meng, Z. C. Yang, and B. Tian,J. Math. Anal. Appl. 336, 1305 (2007); J. Li, H. Q.Zhang, T. Xu, Y. X. Zhang, and B. Tian, J. Phys. A 40,13299 (2007); T. Xu, C. Y. Zhang, G. M. Wei, J. Li,X. H. Meng, and B. Tian, Eur. Phys. J. B 55, 323(2007); W. J. Liu, X. H. Meng, K. J. Cai, X. Lü, T. Xu,and B. Tian, J. Mod. Opt. 55, 1331 (2008).

[7] I. Hacinliyan and S. Erbay, Chaos, Solitons and Frac-tals 14, 1127 (2002).

[8] T. Kanna, M. Lakshmanan, P. Tchofo Dinda, and NailAkhmediev, Phys. Rev. E 73, 026604 (2006).

[9] S. V. Manakov, Sov. Phys. JETP 38, 248 (1974); J. E.Rothenberg, Opt. Lett. 16, 18 (1991).

[10] P. K. A. Wai, C. R. Menyuk, and H. H. Chen, Opt.Lett. 16, 1231 (1991); C. De Angelis, S. Wabnitz, andM. Haelterman, Electron. Lett. 29, 1568 (1993); H. Q.Zhang, X. H. Meng, T. Xu, L. L. Li, and B. Tian, Phys.Scr. 75, 537 (2007).

[11] T. Kanna and M. Lakshmanan, Phys. Rev. E 67,046617 (2003).

[12] C. R. Menyuk, IEEE J. Quant. Elec. 23, 174 (1987).[13] M. Wadati, T. Izuka, and M. Hisakado, J. Phys. Soc.

Jpn. 61, 2241 (1992); M. Lakshmanan, T. Kanna, andR. Radhakrishnan, Rep. Math. Phys. 46, 143 (2000);G. K. Newboult, D. F. Parker, and T. R. Faulkner,J. Math. Phys. 30, 930 (1989).

[14] G. P. Agrawal, Nonlinear Fiber Optics, AcademicPress, New York 1995.

[15] R. Sahadevan, K. M. Tamizhmani, and M. Laksh-manan, J. Phys. A 19, 1783 (1986).

[16] K. Nakkeeran, J. Mod. Opt. 48, 1863 (2001).[17] Q. H. Park and H. J. Shin, Phys. Rev. E 59, 2372

(1999).[18] H. Q. Zhang, J. Li, T. Xu, Y. X. Zhang, W. Hu, and

B. Tian, Phys. Scr. 76, 452 (2007).[19] C. R. Menyuk, IEEE J. Quant. Elec. 25, 2674 (1989).[20] D. N. Christodoulides and M. I. Carvalho, J. Opt. Soc.

Am. B 12, 1628 (1995); A. Ankiewicz, W. Kro-likowski, and N. N. Akhmediev, Phys. Rev. E 59, 6079(1999).

[21] D. N. Christodoulides and R. I. Joseph, Opt. Lett.13, 53 (1988); M. V. Tratnik and J. E. Sipe, Phys.Rev. A 38, 2011 (1988); N. Akhmediev, W. Kro-likowski, and A. W. Snyder, Phys. Rev. Lett. 81, 4632(1998).

[22] D. N. Christodoulides, T. H. Coskun, and R. I. Joseph,Opt. Lett. 22, 1080 (1997); D. N. Christodoulides,T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev.Lett. 78, 646 (1997); W. Krolikowski, N. Akhme-diev, and B. Luther-Davies, Phys. Rev. E 59, 4654(1999).

[23] R. Hirota, Phys. Rev. Lett. 27, 1192 (1971); Prog.Theor. Phys. Suppl. 52, 1498 (1974); R. Hirota andJ. Satsuma, J. Phys. Soc. Jpn. 41, 2141 (1976); Y. Mat-suno, Bilinear Transformation Methods, AcademicPress, New York 1984; R. Hirota, The Direct Methodin Soliton Theory, Cambridge University Press, Cam-bridge 2004.