Bright N-Soliton Solutions to the Vector Hirota Equation from Nonlinear Optics with Symbolic Computation

Bright N-Soliton Solutions to the Vector Hirota Equation from Nonlinear Optics with Symbolic Computation

Tao Xu^a,b, Bo Tian^a,b,c, and Feng-Hua Qi^a,b

a State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

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

c State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China

Reprint requests to B. T.; E-mail:tian.bupt@yahoo.com.cn

Z. Naturforsch.67a,39 – 49 (2012) / DOI: 10.5560/ZNA.2011-0055 Received March 9, 2011 / revised July 17, 2011

Under investigation in this paper is the vector Hirota (VH) equation which governs the simulta- neous propagation of multiple interacting femtosecond pulses in a certain type of coupled optical waveguides. By theNth iterated Darboux transformation starting from the zero potential, the VH equation is found to admit the brightN-soliton solutions in terms of the multi-component Wron- skian. Asymptotic formulae of the brightN-soliton solutions are derived for any given set of spectral parameters, which allows us to directly analyze the collision dynamics of VH solitons. Via symbolic computation, some collision properties possessed by the two- and three-soliton solutions are revealed from four aspects: the asymptotic patterns of the colliding solitons, parametric conditions for the amplitude-preserving collisions, phase shifts induced by the vector-soliton collisions, and soliton state changes described by the generalized linear fractional transformations.

Key words:Vector Solitons; Vector Hirota Equation; Multi-Component Wronskian; Darboux Transformation; Symbolic Computation.

PACS numbers:05.45.Yv; 02.30.Ik

1. Introduction

Vector solitons (VSs) consist of two or more com- ponents (modes) that mutually self-trap in a nonlinear medium [1]. Manakov [2] has firstly suggested VSs comprised of two orthogonally polarized components in a nonlinear Kerr medium under the assumption that the self-phase modulation is identical to the cross- phase modulation. VSs have been experimentally ob- served in planar wave guides [3], birefringent optical fibers [4], photorefractive materials [5], and fiber laser resonators [6]. It has been shown that the VSs can be coupled in different combinations of the bright and dark solitons [7,8]. For example, the two-component VSs admit the bright–bright, dark–dark, and bright–

dark coupled pairs [8].

As a prototype model for the VSs, the vector non- linear Schr¨odinger (VNLS) equation [1,2,9],

iu_z+u_tt+2σ||u||²u=0, u= (u₁,u₂, . . . ,u_m), (1)

is completely integrable [2,9,10] and has physical rel- evance in soliton communication systems [1], opti- cal waveguide systems [1,3], multi-component Bose–

Einstein condensates [11], and so on, where σ=±1 respectively denote the focusing and defocusing cases, u_j represents the jth optical field (1≤ j ≤ m), z andtrespectively denote the direction of propagation and retarded time, and || · || is the Euclidean norm.

Due to the multi-component structure, the VNLS soli- tons with internal degrees of freedom possess more complicated collision properties than the scalar ones although their collisions are also considered to be elas- tic in the sense that the total energy of each collid- ing soliton is conserved [12–14]. Bright VNLS soli- tons can exhibit the amplitude-preserving collisions with no energy exchange among all the components and amplitude-changing collisions along with energy exchange among the components, depending on the pre-collision soliton parameters [12–15]. Recently, amplitude-changing collisions have been observed for

c

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

the spatial Manakov-like solitons in the Kerr and Kerr- like media [16] and temporal ones in the linearly bire- fringent optical fibers [4]. Such collisions have the potential applications in implementing the all-optical digital computation by virtue of some virtual logic gates [16,17].

Based on the work of [12–14,17], an indirect method of analyzing the collisions of bright VNLS solitons can be summarized as follows: (i) AnN-soli- ton collision can be decomposed into N(N−1)/2 pairwise collisions because the multi-soliton collision process has been proved to be pairwise and indepen- dent of the order in which the collisions occur [12];

(ii) As (1) admits an SU(m) symmetry, the pairwise soliton collisions withm>2 components can be re- duced to those in the two-component case by a unitary transformation [14]; (iii) Two-soliton collisions of the two-component VNLS solitons are described by a couple of linear fractional transformations (LFTs), by which the operators defined can form a M¨obius transformation group [17]. On the other hand, it has been found that the brightN-soliton solutions to the fo- cusing VNLS equation can be represented in terms of the multi-component Wronskian [18,19]. Moreover, an algebraic procedure has been derived in [18,19], which can be used to directly analyze the collisions of bright VNLS solitons for any givenN andm. It is emphasized that the method used in [18,19] does not require the collisions of VSs to be pairwise, that is to say, such method can be used to directly analyze the simultaneous collisions among three or more VSs.

In this paper, we would like to apply the method used in [18,19] to a generalized higher-order VNLS equation, i.e., the vector Hirota (VH) equation,

iu_z+1

2u_tt+||u||²u+iεu_ttt+3 iε||u||²u_t +3 iε(u^∗·u_t)u=0, u= (u₁,u₂, . . . ,um),

(2) which can be used to describe the propagation dynam- ics of multiple interacting femtosecond pulses in a cer- tain type of coupled optical waveguides [20], where the parameterε is the ratio of the width of the spectra to the carrier frequency, and the asterisk stands for com- plex conjugate. Ultra-short soliton pulses described by (2) might be capable of increasing the transmission capacity of information systems in the form of the wavelength division multiplexing network which can handle more channels with the minimum frequency difference [1].

Equation (2) is also a completely-integrable model [20–22] and possesses the Painlev´e prop- erty [23], bright N-Soliton solutions [23], bilinear B¨acklund transformation (BT) [24], and Lax pair- based BT [25]. From the perspective of the in- verse scattering transform, [26] has reported the

‘inelastic’¹ two- and three-soliton collisions occur- ring in (2). However, some problems have still not been uncovered for the soliton solutions to (2): (i) What is the determinant representation of the gen- eral bright N-soliton solutions? (ii) Is the method in [18,19] applicable to the collision dynamics of VH solitons? (iii) What are the parametric condi- tions for the occurrence of the amplitude-preserving and amplitude-changing collisions? (iv) Can we ex- plicitly give the formulae for describing the soli- ton phase shifts and state changes in the collision process?

With symbolic computation [27–32], the structure of this paper will be arranged as follows: In Section2, we will convert (2) into the vector complex modi- fied Korteweg-de Vries (VCMKdV) equation and con- struct theNth iterated Darboux transformation (DT).

In Section3, by virtue of Cramer’s rule, we will give the (m+1)-component Wronskian representation of the brightN-soliton solutions to (2). In Section4, we will derive the asymptotic expressions of the brightN- soliton solutions and reveal some properties of the two- and three-soliton collisions. In Section5, we will ad- dress the conclusions.

2.Nth Iterated DT

Through the transformations u(t,z) =q(T,Z)eⁱ

t 6ε− ^z

108ε2

, Z=εz, T =t− z

12ε,

(3) (2) can be transformed into the following VCMKdV equation:

q_Z+q_{T T T}+3||q||²q_T+3(q^∗·q_T)q=0,

q= (q₁,q2, . . . ,q_m), (4)

whose Lax pair is in the (m+1)×(m+1)-matrix form [2,10]

ΨT =U(λ)Ψ= (λU0+U₁)Ψ,

1The authors think it is more correct to use the term ‘amplitude- changing’ instead of ‘inelastic’ because of the vector nature of (2).

ΨZ=V(λ)Ψ= (λ³V0+λ²V1+λV2+V₃)Ψ, (5) with

U0=

1 0 0 −E_m

, U1=

0 q

−q^† 0

, V₀=−4U₀, V₁=−4U₁,

V₂=−2 qq^† q_T q^†_T q^†q

! ,

V₃= qq^†_T−q_Tq^† −q_{T T}−2qq^†q q^†_{T T}+2q^†qq^† q^†q_T−q^†_Tq

! ,

where Ψ = (ψ₁,ψ2, . . . ,ψm+1)^T (T represents the transpose of a vector) is the vector eigenfunction, λ is the spectral parameter,E_mis them×midentity ma- trix, and the dagger denotes the Hermitian conjugate (i.e., transpose and conjugate). One can check that the compatibility conditionΨ_TZ =Ψ_ZT is exactly equiva- lent to (4).

Since the Lax pair of (4) has been obtained, we would like to adopt the DT method [33] to find the determinant representation of soliton solutions. It is known that the DT is such a kind of gauge transfor- mation that leaves the form of a Lax pair invariant, and is in general comprised of the eigenfunction and po- tential transformations [33]. We assume that theNth iterated eigenfunction transformation for (5) be of the form

Ψ[N] =Γ[N](λ)Ψ, (6) in which Ψ[N] = (ψ₁[N],ψ₂[N], . . . ,ψ_m+1[N])^T is the Nth iterated eigenfunction that satisfies ΨT[N]

= U[N](λ)Ψ[N] and ΨZ[N] =V[N](λ)Ψ[N] with U[N](λ) andV[N](λ) being the same as U(λ) and V(λ)except thatqis replaced by theNth iterated po- tential q[N] = (q₁[N], . . . ,q_m[N]), andΓ[N](λ)is the undeterminedNth iterated Darboux matrix,

Γ[N](λ) =















A[N](λ) B₁[N](λ) · · · B_m[N](λ) C₁[N](λ) D₁₁[N](λ) · · · D_1m[N](λ)

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

Cm[N](λ) Dm1[N](λ) · · · Dmm[N](λ)















, (7)

with

A[N](λ) =λ^N−

N−1

∑

n=0

αnλⁿ,

Bj[N](λ) =

N−1 n=0

∑

βjn(−λ)ⁿ, C_i[N](λ) =−

N−1

∑

n=0

γ_inλⁿ,

(8)

D_ii[N](λ) =λ^N+

N−1

∑

n=0

δ_ii⁽ⁿ⁾(−λ)ⁿ, D_{i j}[N](λ) =

N−1 n=0

∑

δ_{i j}⁽ⁿ⁾(−λ)ⁿ (i6=j),

(9)

whereβ_jn,γ_in, andδ_{i j}⁽ⁿ⁾(1≤i,j≤m; 0≤n≤N−1) are the functions ofT andZto be determined.

Suppose thatΨ0k = (f_k,g⁽¹⁾_k , . . . ,g^(m)_k )^Tis the gen- eral solution of (5) which corresponds toλ =λ_k(1≤ k≤N). We generate a sequence of vector functions {Ψ_jk}^m_j=1 which are orthogonal to Ψ_0k, with Ψ_jk = (−g⁽_k^j)∗,

j−1

z }| { 0, . . . ,0,f_k^∗,

m−j

z }| {

0, . . . ,0)^T for 1≤ j ≤m. Since {Ψ_0k}^N_k=1are linearly independent of one another, the functionsα_n,β_jn,γ_in, andδ_{i j}⁽ⁿ⁾(1≤i,j≤m; 0≤n≤ N−1) can be uniquely determined by requiring that

Γ[N](λ_k)Ψ_0k=0, Γ[N](−λ_k^∗)Ψ_jk=0

(1≤k≤N; 1≤j≤m), (10) which can be expanded as

f_kA[N](λ_k) +

m n=1

∑

g⁽ⁿ⁾_k B_n[N](λ_k) =0, f_k^∗B_j[N](−λ_k^∗)−A[N](−λ_k^∗)g^(j)∗_k =0,

(11)

f_kC_i[N](λ_k) +

m

∑

n=1

g⁽ⁿ⁾_k D_in(λ_k) =0, f_k^∗D_{i j}[N](−λ_k^∗)−C_i[N](−λ_k^∗)g⁽_k^j)∗=0,

(12)

where 1≤i,j≤mand 1≤k≤N.

With theNth iterated Darboux matrixΓ[N](λ)de- termined by (11) and (12), one can verify that the spatial- and temporal-flow invariant conditions ΓT[N](λ) +Γ[N](λ)U(λ) =U[N](λ)Γ[N](λ), (13a) ΓZ[N](λ) +Γ[N](λ)V(λ) =V[N](λ)Γ[N](λ), (13b)

are satisfied under theNth iterated potential transfor- mations:

q[N] =q+2(−1)^NbN−1, q^∗[N] =q^∗+2cN−1, (14) where b_N−1 = (β_1,N−1, . . . ,βm,N−1) and c_N−1 = (γ1,N−1, . . . ,γm,N−1) with (−1)^Nβj,N−1 =γ^∗_j,N−1 for 1≤j≤m.

Therefore, (6) and (14) constitute theNth iterated DT (Ψ,q)→ (Ψ[N],q[N]) for (4) [or equivalently, (2)]. We omit the proof of (13a) and (13b) because the focus of this paper is to analyze the collision dynamics of VH solitons.

3. BrightN-Soliton Solutions in Terms of the (m+++1)-Component Wronskian

From (11) and (12), Cramer’s rule enables us to ob- tainβj,N−1andγj,N−1(1≤ j≤m) as

βj,N−1= (−1)^jN−1χ_j τ , γj,N−1= (−1)^(j−1)N−1χ¯_j

τ ,

(15)

with τ, χj, and ¯χj being the following (m+1)- component Wronskians:

τ=

F_N −G⁽¹⁾_N · · · −G⁽_N^j) · · · −G^(m)_N G^(1)∗_N F_N^∗ · · · 0 · · · 0

... ... . .. ... . .. ... G⁽_N^j)∗ 0 · · · F_N^∗ · · · 0 ... ... . .. ... . .. ... G^(m)∗_N 0 · · · 0 · · · F_N^∗

,(16)

χ_j=

F_N+1 −G⁽¹⁾_N · · · −G⁽_N−1^j) · · · −G^(m)_N G^(1)∗_N+1 F_N^∗ · · · 0 · · · 0

... ... . .. ... . .. ... G⁽_N+1^j)∗ 0 · · · F_N−1^∗ · · · 0 ... ... . .. ... . .. ... G^(m)∗_N+1 0 · · · 0 · · · F_N^∗

,

(17) χ¯_j=

FN−1 −G⁽¹⁾_N · · · −G⁽_N+1^j) · · · −G^(m)_N G^(1)∗_N−1 F_N^∗ · · · 0 · · · 0

... ... . .. ... . .. ... G⁽_N−1^j)∗ 0 · · · F_N+1^∗ · · · 0 ... ... . .. ... . .. ... G^(m)∗_N−1 0 · · · 0 · · · F_N^∗

,

(18) where the block matricesFMandG⁽_M^j)(1≤j≤m;M= N−1,N,N+1)are given by

FM=















f1 ∂f₁

∂T · · · ^∂M−1f1

∂T^M−1

f₂ ^∂_∂T^f2 · · · ^∂M−1f2

∂T^M−1

... ... . .. ... fN ∂f_N

∂T · · · ^∂M−1fN

∂T^M−1













 ,

G⁽_M^j)=



















g⁽₁^{j) ∂g}

(j) 1

∂T · · · ^∂M−1∂g

(j) 1

∂T^M−1

g⁽₂^{j) ∂g}

(j) 2

∂T · · · ^∂M−1∂g

(j) 2

∂T^M−1

... ... . .. ... g⁽_N^{j) ∂g}

(j) N

∂T · · · ^∂

M−1∂g^(j)_N

∂T^M−1

















 .

(19)

With regard to the multi-component Wronskians τ, χj, and ¯χj, we make the following two re- marks: First, the function τ is a complex-valued one which can be expanded asτ=ϒ(m)τ⁰ withϒ(m) = e^{∑Nk=1(m−1)θk∗} ∏

1≤k<l≤N

(λ_l^∗−λ_k^∗)^m−1, whereτ⁰ is a real- valued function and has no zeros for all(Z,T)∈R². That is to say, the solutionq[N]and its complex con- jugateq^∗[N]have no singularity in theZT plane, pro- vided thatλ_k6=λ_lfor 1≤k<l≤N. Second,χ_j/ϒ(m) is complex conjugate to ¯χj/ϒ(m), which implies that (−1)^Nβj,N−1=γ^∗_j,N−1(1≤ j≤m). We will prove the above two facts by the Laplace expansion theorem and Binet–Cauchy theorem in a separate paper.

Withq=0andλ =λ_k (1≤k≤N), the solutions of (5) can be given as

(f_k,g⁽¹⁾_k ,g⁽²⁾_k , . . . ,g^(m)_k ) =

(a_ke^θk,b⁽¹⁾_k e^−θk,b⁽²⁾_k e^−θk, . . . ,b^(m)_k e^−θk), (20)

with θ_k=λ_kT−4λ_k³Z, a_k and b⁽_k^j) (1≤ j≤m) as complex constants. Thus, the first transformation in (14) yields theNth iterated solutionq[N]to (4) in the (m+1)-component Wronskian form

q[N] =−2 τ

χ₁, . . . ,(−1)^(j+1)Nχ_j, . . . ,(−1)^(m+1)Nχm

.

(21)

For the caseN=1, (21) can be expressed as q[1] =|a₁|κ₁B^∗₁

a^∗₁||B₁|| e(^λ1−λ₁^∗)T−4(λ₁³−λ₁^∗3)Z

·sech

κ1T−ω1Z+ln |a₁|

||B₁||

,

(22)

which is called the bright one-soliton solutions ifa16=

0 and B₁6=0, whereB₁= (b⁽¹⁾₁ ,b⁽²⁾₁ , . . . ,b^(m)₁ ),κ₁= λ₁+λ₁^∗is the wave number,ω₁=4

λ₁³+λ₁^∗3 is the frequency, ¹

κ1ln_||B^|a1|

1||is the initial phase, and the soliton amplitude and velocity are respectively given by

A₁= |κ₁|

||B₁|| |b⁽¹⁾₁ |,|b⁽²⁾₁ |, . . . ,|b^(m)₁ |T

, v₁=4 λ₁²− |λ₁|²+λ₁^∗2

.

(23)

ForN≥2, (21) can describe the collision dynam- ics of the brightN-soliton solutions excluding one sin- gular situationa_k=0 and four reducible situations: (i) λ_k=0; (ii)λ_k=λ_l^∗; (iii)B_k= (b⁽¹⁾_k ,b⁽²⁾_k , . . . ,b^(m)_k ) =0;

(iv)λ_k²+λ_k^∗2− |λ_k|²=λ_l²+λ_l^∗2− |λ_l|²(k6=l). Here, the word ‘reducible’ means that the N-soliton solu- tions will be reduced to the(N−1)-soliton solutions for the first three cases, and to the partially coherent solitons [34,35] or multi-soliton complexes [36] for the last case (see detailed analysis in [18,19]). More- over, we can without loss of generality takea_k=1 for 1≤k≤N in (21), which implies that the bright N- soliton solutions to (4) [or equivalently, (2)] are char- acterized by(m+1)Ncomplex parameters:λkandb⁽_k^j) (1≤k≤N; 1≤j≤m). We note that the bright soliton solutions obtained by the Hirota method [23], the ones in terms of the double Wronskian [24], and the ones via the Lax pair-based BT [25] contain less than(m+1)N free parameters. Therefore, (21) is more general than those obtained in [23–25].

4. Asymptotic Analysis of the BrightN-Soliton Solutions

Since the brightN-soliton solutions to (2) also admit the (m+1)-component Wronskian representation, in this section we will follow the way in [18,19] to derive the expressions of asymptotic solitons for the bright N-soliton solutions with any given set of the spec- tral parameters{λ_k}^N_k=1. Then, we will use the method in [18,19] to analyze the two- and three-soliton colli- sions. For convenience, throughout this section we de- fine the notations[n]:={1,2, . . . ,n},[k,n]:={k,k+ 1, . . . ,n}, and use| · |to denote the number of elements of a set.

4.1. Asymptotic Formulae

For a given set of the spectral parameters{λ_k}^N_k=1, we assume thatµ_k= (λ_k+λ_k^∗),ν_k=i(λ_k^∗−λ_k) andr_k=4(λ_k²+λ_k^∗2− |λ_k|²) =µ_k²−3ν_k²for 1≤k≤ N. Without loss of generality, the spectral parameters {λ_k}^N_k=1can be ordered according to the relationr₁<

r₂< . . . <r_Nbecause (21) is required to be irreducible.

Before applying the method in [18,19] to (21), we give the following three important properties as the basis for us to derive the asymptotic expressions for this bright N-soliton solutions asZ→ ∓∞.

The first is for the limiting states of Re(θ_k) =

1

2µ_k(T−r_kZ)asZ→ ∓∞for 1≤k≤N. If Re(θ_n)∼0 in theT Z-plane asZ→ ∓∞for somen∈[N], we can make use of the relation

µnRe(θ_k) =µkRe(θ_n) +1

2µnµk(r_n−r_k)Z, (24) to determine that, (i) as Z→ −∞, Re(θ_k)∼ −∞ for k∈ B^(I)_n ∪ B^(II)_n and Re(θ_k)∼+∞fork∈ B^(III)_n ∪ B^(IV)_n ; (ii) asZ→+∞, Re(θ_k)∼+∞fork∈ B^(I)_n ∪ B^(II)_n and Re(θ_k)∼ −∞fork∈ B^(III)_n ∪ B^(IV)_n . Here, the setsB^(I)_n , B^(II)_n , B^(III)_n , and B^(IV)_n are defined as B^(I)_n ={l|µ_l >

0 andl∈[n−1]},B^(II)_n ={l|µ_l<0 andl∈[n+1,N]}, B^(III)_n ={l|µ_l<0 andl∈[n−1]}, andB^(IV)_n ={l|µ_l>

0 andl∈[n+1,N]}.

The second is for the linear relation of phase combi- nations in the expansions ofτandχ_j(1≤j≤m). The expansions ofτandχj(1≤j≤m)are the sums of ex- ponential terms with the exponents respectively as the linear phase combinationsϑτ=∑^N_k=1 c^(I)_k θ_k+d^(I)_k θ_k^∗ and ϑ_χj =∑^N_k=1 c^(II)_k θ_k+d_k^(II)θ_k^∗

, where c^(I)_k ,c^(II)_k ∈

{−1,1}, d_k^(I),d^(II)_k ∈ {m−2,m},

{k|c^(I)_k =−1}

= {k|d_k^(I) =m−2}

,

{k|c^(I)_k =1}

=

{k|d_k^(I) =m}

, {k|c^(II)_k =−1}

=

{k|d_k^(II)=m−2}

−1,

{k|c^(II)_k = 1}

=

{k|d^(II)_k =m}

+1.

The third is for the asymptotically-dominating (AD) behaviour of τ and χj (1≤ j≤m)as Z → ∓∞. If Re(θ_n)∼0 in the T Z-plane as Z → ∓∞ for some n∈[N], then we have e^{ϑτ−ϑτ,lAD}∼0 or O(1) (l=1,2) and e^ϑχj−ϑ

AD

χj ∼0 orO(1) as Z→ ∓∞, with Θ_τ,1^∓ = θ_n+mθ_n^∗+Ξ_n^∓ andΘ_τ,2^∓ = (m−2)θ_n^∗−θ_n+Ξ_n^∓ as the linear phase combinations associated with the AD terms in the expansion of τ as Z→ ∓∞, andΘ_χ^∓

j =

θn+ (m−2)θ_n^∗+Ξ_n^∓as the one associated with the AD term in the expansion of χ_j(1≤ j≤m)asZ→ ∓∞, whereΞ_n^∓= ∑

k6=n

(m−1)θ_k^∗+σ_kn^∓(θ_k+θ_k^∗)

, andσ_kn^∓’s are given by

σ_kn⁻=







−1, fork∈ B^(I)_n ∪ B^(II)_n , 1, fork∈ B^(III)_n ∪ B^(IV)_n , 0, fork=n,

σ_kn⁺=







1, fork∈ B^(I)n ∪ B^(II)n ,

−1, fork∈ B^(III)_n ∪ B^(IV)_n , 0, fork=n.

(25)

Therefore, for any given set of the spectral parame- ters{λ_k}^N_k=1, we can obtain the expressions for thenth asymptotic soliton (1≤n≤N) of (21) asZ→ ∓∞as follows:

S^∓_n = (s^∓_1n, . . . ,s^∓_jn, . . . ,s^∓_mn)

= −e^θn−θn∗ 2p

c^∓_nd^∓_n

(26)

·

e^∓_1n, . . . ,(−1)^(j+1)Ne^∓_jn, . . . ,(−1)^(m+1)Ne^∓_mn

·sech



θn+θ_n^∗+ln s

c^∓_n d_n^∓



,

wherec^∓_n andd_n^∓respectively correspond to the coeffi- cients of AD terms associated withΘ_τ,1^∓ andΘ_τ,2^∓ in the expansion ofτ, ande^∓_{n j}is the coefficient of AD term as- sociated withΘ_χ^∓_j in the expansion ofχj(1≤j≤m).

4.2. Two-Soliton Collisions

For the two-soliton collisions described by (21) with N =2, we can employ (26) to derive two different

asymptotic expressions for the nth colliding soliton (n=1,2) as follows:

S⁽ⁱ⁾_n =µnA⁽ⁱ⁾_n

||A⁽ⁱ⁾_n ||

e^θn−θn∗sech

µn(T−r_nZ) +ln∆_n⁽ⁱ⁾ (i=1,2),

(27) with

∆_n⁽¹⁾=|λ_n−λ3−n|²

||A⁽¹⁾_n || ,

∆n⁽²⁾=|λ3−n−λ_n^∗|²||B3−n||²

||A⁽²⁾_n ||

,

A⁽¹⁾_n = a⁽¹⁾_1n, . . . ,a⁽¹⁾mn

= (λ_n−λ_3−n) (λ_3−n−λ_n^∗)B^∗_n, A⁽²⁾_n = a⁽²⁾_1n, . . . ,a⁽²⁾mn

= λ_3−n^∗ −λn

·

(λ_n^∗−λ3−n)||B3−n||²B^∗_n + (λ3−n−λ_3−n^∗ )(B^∗_n·B3−n)B^∗_3−n

,

where the superscript ‘(i)’ represents the state of each asymptotic soliton, and the dot denotes the product of vectors.

It is easy to find thatA⁽ⁱ⁾_n and∆n⁽ⁱ⁾(1≤n≤2; 1≤i≤ 2) in (27) are exactly the same as those in the asymp- totic expressions of the two-soliton solutions to the focusing VNLS equation [19]. Accordingly, the two- soliton collisions described by (21) with N =2 also admit the following properties:

(i) According to the signs ofµ1 andµ2, (21) with N=2 is found to admit four kinds of asymptotic pat- terns as Z → ∓∞, which are shown in Table1. Ve- locities and total energies of each colliding soliton are the same before and after collision, i.e., v⁻_n = v⁺_n =r_n,E_n⁻=E_n⁺=2|µ_n|. The parametric conditions r1r2>0 andr1r2<0, respectively, correspond to the overtaking and head-on collisions between two soli- tons.

(ii) Either

B₁kB₂ or B₁⊥B^∗₂, (28) Table 1. Asymptotic patterns of (21) withN=2 under four different parametric conditions.

Parametric Asymptotic solitons Asymptotic solitons

conditions (Z→ −∞) (Z→+∞)

µ1>0,µ2>0 S⁻₁ =S⁽¹⁾₁ ,S⁻₂ =S⁽²⁾₂ S⁺₁ =S⁽²⁾₁ ,S⁺₂ =S⁽¹⁾₂ µ1<0,µ2>0 S⁻₁ =S⁽¹⁾₁ ,S⁻₂ =S⁽¹⁾₂ S⁺₁ =S⁽²⁾₁ ,S⁺₂ =S⁽²⁾₂ µ1>0,µ2<0 S⁻₁ =S⁽²⁾₁ ,S⁻₂ =S⁽²⁾₂ S⁺₁ =S⁽¹⁾₁ ,S⁺₂ =S⁽¹⁾₂ µ1<0,µ2<0 S⁻₁ =S⁽²⁾₁ ,S⁻₂ =S⁽¹⁾₂ S⁺₁ =S⁽¹⁾₁ ,S⁺₂ =S⁽²⁾₂

3

T

2 2

Z

0 3

q

3

T

3

T

2 2

Z

0 3

q

3

T

(a) (b)

Fig. 1 (colour online). Amplitude-changing collision between two bright vector solitons withm=2, where the related param- eters are chosen asB₁= (1,0),B₂= (−1,1),λ1=−³₂+i, andλ2=−2+i. Note that the second component of the second colliding soliton vanishes after the collision.

being satisfied, the amplitudes for all the components are preserved in the two-soliton collision process; oth- erwise, the amplitudes for some or all components will change after collision along with the energy ex- change among the relevant components (see Figs.1a and1b). It is mentioned that the bright two-soliton so- lutions obtained in [23,24] are subject to the condition B₁kB₂, so they just exhibit the amplitude-preserving

collisions. In particular, the amplitude of the jth com- ponent (1≤ j≤m) for the nth colliding soliton be- comes zero ifa⁽ⁱ⁾_jn=0 forS⁽ⁱ⁾_n (i=1,2), as displayed in Figure1a.

(iii) Given the parametersλ1andλ2, the phase shift of thenth colliding soliton is dependent on the angle φ_n between the vectors B^∗_n and B_3−n in the explicit form

Φ_n^(1↔2) =

ln∆_n⁽²⁾

∆n⁽¹⁾

=

ln |λ_n−λ_3−n^∗ |²

|λ_n−λ3−n|

|λ_n−λ_3−n^∗ |²+ (λ_n−λ_n^∗)(λ3−n−λ_3−n^∗ )|cosφn|²¹₂ ,

(29)

where Φ_n^(1↔2) reaches the maximum or minimum value atφn=0 (B^∗_nkB_3−n) orφn=^π₂ (B^∗_n⊥B_3−n).

(iv) We use the vectorsρ_n⁽ⁱ⁾= ρ_1n⁽ⁱ⁾,ρ_2n⁽ⁱ⁾, . . . ,ρ_mn⁽ⁱ⁾

=

1,^a

(i) 2n

a⁽ⁱ⁾_1n, . . . ,^a(i)mn

a⁽ⁱ⁾_1n

(i=1,2)to stand for the two asymp- totic states of the nth colliding soliton as Z → ∓∞.

Change of thenth colliding soliton from the state ‘1’

to ‘2’ can be described by the following generalized LFTs:

ρ_n⁽²⁾ =T^(1→2)_n ρ_n⁽¹⁾

=

ρ_3−n⁽¹⁾

2ρn⁽¹⁾+hn ρ_3−n^(1)∗·ρn⁽¹⁾ ρ_3−n⁽¹⁾

ρ_3−n⁽¹⁾

2+hn ρ_3−n^(1)∗·ρn⁽¹⁾

h_n=λ3−n−λ_3−n^∗ λ_n^∗−λ3−n

,

(30)

where the operatorsT^(1→2)_n (n=1,2) directly reflect the state changes undergone by two colliding solitons which contain m components, without reducing the m-component soliton collisions to the two-component ones by a unitary transformation[14].