On the Nth Iterated Darboux Transformation and Soliton Solutions of a Coherently-Coupled Nonlinear Schr¨odinger System

On the Nth Iterated Darboux Transformation and Soliton Solutions of a Coherently-Coupled Nonlinear Schr¨odinger System

Tao Xu^a, Ping-Ping Xin^a, Yi Zhang^a, and Juan Li^b,c

a College of Science, China University of Petroleum, Beijing, 102249, China

b State Key Laboratory of Remote Sensing Science, Jointly Sponsored by the Institute of Remote Sensing Applications of Chinese Academy of Sciences, and Beijing Normal University, Beijing 100101, China

c Demonstration Centre, Spaceborne Remote Sensing National Space Administration, Beijing 100101, China

Reprint requests to T. X.; E-mail:xutao@cup.edu.cn

Z. Naturforsch.68a,261 – 271 (2013) / DOI: 10.5560/ZNA.2012-0110

Received June 8, 2012/ revised October 7, 2012 / published online February 6, 2013

In this paper, we study an integrable coherently-coupled nonlinear Schr¨odinger system arising from low birefringent fibers and weakly anisotropic media. We construct theNth iterated Darboux transformation (DT) in the explicit form and give a complete proof for the gauge equivalence of the associated Lax pair. By the DT-based algorithm, we derive theN-soliton solutions which can be uniformly represented in terms of the four-component Wronskians. We analyze the properties of coherently coupled solitons, revealing the parametric criterion for the non-degenerate solitons to respectively display the one- and double-hump profiles. In addition, we point out that the double- hump solitons may have potential application in realizing the multi-level optical communication.

Key words:Coherently-Coupled Nonlinear Schr¨odinger System; Darboux Transformation; Soliton Solutions; Four-Component Wronskians.

PACS numbers:05.45.Yv; 02.30.Ik

1. Introduction

Coupled (or symbiotic) solitons with two or more components (modes) [1] have been experimentally ob- served in the AlGaAs planar wave guides [2], birefrin- gent optical fibers [3], photorefractive materials [4], and fiber laser resonators [5]. In the Kerr or Kerr-like media, the co-propagation of two optical fields is usu- ally governed by the coupled nonlinear Schr¨odinger (NLS) system [6],

iq_j,z+q_j,tt+2 |q_j|²+|q3−j|² q_j=0

(j=1,2),(1) which is also known as the Manakov system [7]. Equa- tion (1) is said to be incoherently because the cou- pling is phase insensitive [8]. One of the most attrac- tive properties associated with (1) is the Manakov soli- ton collision [3,9–19]. Due to the intensity-coupling structure in (1), the collisions of the Manakov solitons with the internal degrees of freedom are more compli- cated than those for the scalar ones [9–14]. Depending

on the pre-collision soliton parameters, (1) can exhibit the shape-changing collisions along with energy redis- tribution between two components, as well as shape- preserving collisions without energy transfer between two components [9–14]. Such two kinds of collisions are both considered to be elastic in the sense that the total energy of each coupled soliton is conserved [14].

It should be mentioned that the shape-changing colli- sions have been experimentally observed for the spatial coupled solitons in the Kerr-like media [15] and tem- poral ones in the linearly birefringent optical fibers [3].

More importantly, the Manakov soliton collisions have brought about the potential applications in implement- ing the all-optical digital computation [16,17] and de- signing the ‘solitonets’ which are complex networks made up of interacting fields [18,19].

In low birefringent fibers or weakly anisotropic me- dia, one has to take into account the coherent coupling between two optical fields (that is, the coupling is de- pendent on relative phases of the interacting fields [8]).

In this case, the propagation of two optical fields in

© 2013 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

a Kerr-type nonlinear medium is described by the fol- lowing coherently-coupled NLS system [20,21]:

iqj,z+qj,tt+2 |q_j|²+2|q_3−j|²

qj−2 ¯qjq²_3−j=0 (j=1,2),(2) where the bar means complex conjugate,t represents the retarded time for the temporal case or transverse direction for the spatial case,zthe propagation direc- tion,q_j(j=1,2) are slowly varying envelopes of two in- teracting optical fields, the terms|q_j|²q_j,|q3−j|²q_jand

¯

q_jq²_3−j are responsible for the self-phase modulation, cross-phase modulation, and coherent coupling for the energy exchange between two fields, respectively.

As an integrable model [21], (2) governs the prop- agation of optical beams in nonlinear Kerr media with linear optical activity and cubic anisotropy [20], and the trapping of two orthogonally polarized opti- cal pulses in an isotropic medium with the three com- ponents χ_xxyy⁽³⁾, χ_xyxy⁽³⁾, and χ_xyyx⁽³⁾ of the third-order sus- ceptibility tensor χ⁽³⁾ subject to the relation χxxyy⁽³⁾ + χ_xyxy⁽³⁾ =−2χ_xyyx⁽³⁾ [21]. It has been shown that (2) ad- mits degenerate and non-degenerate solitons, where the former is of the usual sech profile [22–25], while the latter contains more free parameters and can dis- play both the one- and double-hump profiles [24–27].

In the optical communication lines, the binary data

‘1’ and ‘0’ can be respectively represented by the presence and absence of an optical soliton, and thus the proximity of individual solitons determines the bit rate of a fiber communication system [6]. As a kind of complex objects formed by the superpo- sition of two fundamental solitons, the double-hump solitons may be appropriate candidates for the multi- level communication in the birefringent or two-mode fibers [28].

In this paper, we will study (2) from the following three aspects: (i) Authors of [26,27] have constructed the once-iterated DT and given the general scheme of Nth iterated DT. However, the explicit form of theNth iterated DT as well as its rigorous proof was absent in [26,27], and the uniform determinantal representa- tion ofN-soliton solutions has also not been obtained.

In the way of [29], we will explicitly construct the Nth iterated DT for (2) and represent the general N- soliton solutions in terms of the four-component Wron- skians. (ii) To our knowledge, the parametric condi- tions under which the one- and double-hump solitons can be respectively generated are still uncovered. Via

the extreme value analysis, we will study the properties of coherently coupled solitons in (2). (iii) The shape- changing collisions of coupled solitons have potential applications in virtual digital computation [16,17] and all-optical switching [30]. Authors of [24,25] have re- ported the shape-changing collisions between degen- erate and non-degenerate solitons. We will explore whether such nontrivial collisions can occur between two degenerate (or non-degenerate) solitons.

2.Nth Iterated Darboux Transformation

In this section, we will construct theNth iterated DT in the explicit form for (2) and give a complete proof for the gauge equivalence of the Lax pair associated with (2).

In the frame of the 4×4 Ablowitz–Kaup–Newell–

Segur inverse scattering formulation [31], the Lax pair of (2) takes the form [21]

Ψt=U(λ)Ψ= (λU^(I)+U^(II))Ψ,

Ψ_z=V(λ)Ψ= (λ²V^(I)+λV^(II)+V^(III))Ψ, (3) with

U^(I)=i

−I 0

0 I

, U^(II)=

0 Q

−Q^† 0

,

V^(I)=2U^(I), V^(II)=2U^(II), V^(III)=i

QQ^† Q_t Q^†_t −Q^†Q

, Q=

q1 q2

−q₂ q₁

,

whereIis the 2×2 unit matrix,Ψ= (ψ₁,ψ2,ψ3,ψ4)^T (the superscript T signifies the vector transpose) is the four-dimensional vector eigenfunction,λ is the spec- tral parameter, and the compatibility conditionU_z(λ)−

V_t(λ) + [U(λ),V(λ) ] =0 is exactly equivalent to (2).

We assume theNth iterated eigenfunction transfor- mation for Lax pair (3) be of the form

Ψ_N=T_N(λ)Ψ, (4)

in whichΨ_N = (ψ_1N,ψ_2N,ψ_3N,ψ_4N)^T is the Nth it- erated eigenfunction that satisfiesΨN,t =UN(λ)ΨN

andΨN,z=V_N(λ)Ψ_N withU_N(λ)andV_N(λ)being the same asU(λ)andV(λ)except thatq1andq2are re- spectively replaced by theNth iterated potentialsq_1N andq_2N, and T_N(λ)is the undeterminedNth iterated

Darboux matrix

T_N(λ) =









A₁₁(λ) A₁₂(λ) B₁₁(λ) B₁₂(λ) A₂₁(λ) A₂₂(λ) B₂₁(λ) B₂₂(λ) C₁₁(λ) C₁₂(λ) D₁₁(λ) D₁₂(λ) C21(λ) C22(λ) C21(λ) D22(λ)







 (5)

with

A_{i j}(λ) = (−iλ)^Nδ_{i j}−

N n=1

∑

a⁽ⁿ⁾_{i j} (−iλ)ⁿ⁻¹

(1≤i,j≤2), (6)

B_{i j}(λ) =−

N

∑

n=1

b⁽ⁿ⁾_{i j} (iλ)ⁿ⁻¹ (1≤i,j≤2), (7) C_{i j}(λ) =−

N

∑

n=1

c⁽ⁿ⁾_{i j} (−iλ)ⁿ⁻¹ (1≤i,j≤2), (8) D_{i j}(λ) = (−iλ)^Nδi j−

N n=1

∑

d_{i j}⁽ⁿ⁾(iλ)ⁿ⁻¹

(1≤i,j≤2), (9)

where δi j is the Kronecker delta function, a⁽ⁿ⁾_{i j} , b⁽ⁿ⁾_{i j} , c⁽ⁿ⁾_{i j} , andd_{i j}⁽ⁿ⁾(1≤i,j≤2; 1≤n≤N) are the functions to be determined.

Note that if Ψ_k = e_k,f_k,g_k,h_kT

satisfies Lax pair (3) with λ = λk (1 ≤ k ≤ N), then Ψ_k⁰ =

f_k,−e_k,h_k,−g_kT

is also a solution of Lax pair (3) withλ =λ_k, andΦ_k= −g¯_k,h¯_k,e¯_k,−f¯_kT

andΦ_k⁰= h¯_k,g¯_k,−f¯_k, −¯e_kT

are the solutions of Lax pair (3) with λ =λ¯_k [26,27]. On the other hand, {Ψ_k}^N_k=1, {Ψ_k⁰}^N_k=1, {Φ_k}^N_k=1, and {Φ_k⁰}^N_k=1 are four sets of linearly-independent solutions. Hence, the functions a⁽ⁿ⁾_{i j} ,b⁽ⁿ⁾_{i j} ,c⁽ⁿ⁾_{i j} , andd_{i j}⁽ⁿ⁾(1≤i,j≤2; 1≤n≤N) can be uniquely determined by requiring that

T_N(λ_k)Ψ_k=0 (1≤k≤N), (10a) TN(λ_k)Ψ_k⁰=0 (1≤k≤N), (10b) T_N(λ¯_k)Φ_k=0 (1≤k≤N), (10c) T_N(λ¯_k)Φ_k⁰=0 (1≤k≤N). (10d) For convenience of calculatinga⁽ⁿ⁾_{i j} ,b⁽ⁿ⁾_{i j} ,c⁽ⁿ⁾_{i j} , and d_{i j}⁽ⁿ⁾ (1≤i,j≤2; 1≤n≤N) from (10a) – (10d) via Cramer’s rule, we introduce the following matrices and

vectors:

E_M=







e₁ · · · (−iλ₁)^M−1e₁ ... . .. ... eN · · · (−iλ_N)^M−1eN





,

F_M=







f₁ · · · (−iλ₁)^M−1f₁ ... . .. ... f_N · · · (−iλ_N)^M−1f_N





,

(11)

G_M=







g1 · · · (iλ₁)^M−1g1

... . .. ... g_N · · · (iλ_N)^M−1g_N





,

H_M=







h₁ · · · (iλ₁)^M−1h₁ ... . .. ... h_N · · · (iλ_N)^M−1h_N





,

(12)

e=

(−iλ₁)^Ne₁, . . .,(−iλ_N)^Ne_N , f=

(−iλ₁)^Nf₁, . . .,(−iλ_N)^Nf_N

, (13)

g=

(iλ₁)^Ng₁, . . .,(iλ_N)^Ng_N , h=

(iλ₁)^Nh₁, . . .,(iλ_N)^Nh_N

, (14)

a_{i j}= a⁽¹⁾_{i j} , . . .,a^(N)_{i j} , b_{i j}= b⁽¹⁾_{i j} , . . .,b^(N)_{i j}

,

(15) c_{i j}= c⁽¹⁾_{i j} , . . .,c^(N_{i j}⁾

, d_{i j}= d_{i j}⁽¹⁾, . . .,d_{i j}^(N) .

(16) Hereby, (10a) – (10d), with the change of the order of equations, can be written in the matrix form

Aτ(a₁₁,a₁₂,b₁₁,b₁₂)^T= (e,f,−¯g,h)¯ ^T, (17a) A_τ(a₂₁,a₂₂,b₂₁,b₂₂)^T= (f,−e,h,¯ g)¯ ^T, (17b) A_τ(c₁₁,c₁₂,d₁₁,d₁₂)^T=

(−1)^N(g,h,e,¯ −¯f)^T, (17c) Aτ(c₂₁,c₂₂,d₂₁,d₂₂)^T=

(−1)^N(h,−g,−¯f,−¯e)^T, (17d) with

A_τ=









E_N F_N G_N H_N F_N −E_N H_N −G_N

−G¯_N H¯_N E¯_N −F¯_N H¯N G¯N −F¯N −E¯N









. (18)

By Cramer’s rule, one can obtain from (17a) – (17d) that

b^(N₁₁⁾=−χ₁₁ τ

, b^(N)₁₂ =(−1)^N−1χ12

τ

, (19)

b^(N₂₁⁾=(−1)^N−1χ21

τ , b^(N)₂₂ =−χ₂₂

τ , (20) c^(N)₁₁ =(−1)^Nχ₁₁⁰

τ , c^(N₁₂⁾=χ₁₂⁰

τ , (21) c^(N)₂₁ = χ₂₁⁰

τ , c^(N)₂₂ =(−1)^Nχ₂₂⁰

τ , (22)

whereτ=|Aτ|;χ_{i j} andχ_{i j}⁰ (1≤i,j≤2) are the fol- lowing determinants:

χ11=

E_N+1 F_N GN−1 H_N FN+1 −E_N HN−1 −G_N

−G¯_N+1 H¯_N E¯N−1 −F¯_N H¯N+1 G¯_N −F¯N−1 −E¯_N ,

χ12=

EN+1 FN GN HN−1

F_N+1 −E_N H_N −GN−1

−G¯N+1 H¯_N E¯_N −F¯N−1

H¯_N+1 G¯_N −F¯_N −E¯N−1

,

(23)

χ21=

E_N F_N+1 GN−1 H_N F_N −EN+1 HN−1 −G_N

−G¯_N H¯_N+1 E¯N−1 −F¯_N H¯_N G¯N+1 −F¯N−1 −E¯_N ,

χ₂₂=

E_N F_N+1 G_N HN−1

F_N −E_N+1 H_N −GN−1

−G¯N H¯N+1 E¯N −F¯N−1

H¯_N G¯_N+1 −F¯_N −E¯N−1

,

(24)

χ₁₁⁰ =

EN−1 FN GN+1 HN

FN−1 −E_N H_N+1 −G_N

−G¯N−1 H¯_N E¯N+1 −F¯_N H¯N−1 G¯_N −F¯_N+1 −E¯_N ,

χ₁₂⁰ =

E_N FN−1 GN+1 H_N F_N −EN−1 H_N+1 −G_N

−G¯_N H¯N−1 E¯_N+1 −F¯_N H¯N G¯N−1 −F¯N+1 −E¯N

,

(25)

χ₂₁⁰ =

EN−1 F_N G_N H_N+1 FN−1 −E_N HN −G_N+1

−G¯N−1 H¯_N E¯_N −F¯_N+1 H¯N−1 G¯_N −F¯_N −E¯N+1

,

χ₂₂⁰ =

E_N FN−1 G_N H_N+1 F_N −EN−1 H_N −GN+1

−G¯_N H¯N−1 E¯_N −F¯_N+1 H¯_N G¯N−1 −F¯_N −E¯_N+1 .

(26)

To verify the form-invariance of Lax pair (3), we need to prove that

TN,t(λ) +TN(λ)U(λ) =UN(λ)T_N(λ), (27a) TN,z(λ) +T_N(λ)V(λ) =V_N(λ)T_N(λ) (27b) are satisfied with the Darboux matrix T_N(λ) given by (5), in whicha⁽ⁿ⁾_{i j} ,b⁽ⁿ⁾_{i j} ,c⁽ⁿ⁾_{i j} , andd_{i j}⁽ⁿ⁾(1≤i,j≤2;

1≤n≤N) are determined by (17a) – (17d). Based on Lemmas1–3(seeAppendix), we can arrive at the fol- lowing proposition:

Proposition 1. Suppose that e_k,f_k,g_k,h_kT

is the so- lution of Lax pair (3) withλ =λ_k(1≤k≤N). Then, the Darboux matrix T_N(λ) in (5) obeys the condi- tions in (27a) and (27b), provided thata⁽ⁿ⁾_{i j} ,b⁽ⁿ⁾_{i j} ,c⁽ⁿ⁾_{i j} , and d_{i j}⁽ⁿ⁾ (1≤i,j ≤2; 1≤n ≤N) are determined by (17a) – (17d), and the Nth iterated potential trans- formations are given by

q1N=q1+2(−1)^N−1b^(N₁₁⁾, q2N=q2+2(−1)^N−1b^(N₁₂⁾.

(28) Proof. We equivalently prove that the Darboux matrix TN(λ)in (5) obeys

U_N(λ)detT_N(λ) =

[T_N,t(λ) +T_N(λ)U(λ)]T_N^∗(λ), (29a) VN(λ)detTN(λ) =

[T_N,z(λ) +TN(λ)V(λ)]T_N^∗(λ), (29b) whereT_N^∗(λ) is the adjoint matrix ofT_N(λ). Let us define that[u_hl(λ)]4×4= [T_N,t(λ) +T_N(λ)U(λ)]T_N^∗(λ) and[v_hl(λ)]4×4= [T_N,z(λ) +T_N(λ)V(λ)]T_N^∗(λ).

By calculation, we have deg[u_hh(λ)] =4N+1,

deg[v_hh(λ)] =4N+2 (1≤h≤4), deg[u_hl(λ)] =4N,

deg[v_hl(λ)] =4N+1 (1≤h,l≤4;h6=l), where deg[f(λ)]represents the degree of the polyno- mial f(λ). On the other hand, Lemmas1and2imply thatu_hl(λ)andv_hl(λ)(1≤h,l≤4) can be exactly di- vided by detT_N(λ). That is to say, the matricesU_N(λ) andV_N(λ)are of the form

U_N(λ) =λU_N^(I)+U_N^(II),

VN(λ) =λ²V_N^(I)+λV_N^(II)+V_N^(III).

(30)

Substituting (30) into (29a) and (29b), and compar- ing the coefficients of λ^4N+1 and λ^4N in (29a) and those ofλ^4N+2,λ^4N+1andλ^4N in (29b), we find that U_N^(I)=U^(I),V_N^(I)=V^(I), andU_N^(II),V_N^(II), andV_N^(III)have the same form asU^(II),V^(II), andV^(III) under the fol- lowing conditions:

q_1N=q₁+2(−1)^N−1b^(N)₁₁ , q_2N=q₂+2(−1)^N−1b^(N)₁₂ ,

(31) q_1N=q₁+2(−1)^N−1b^(N)₂₂ ,

q_2N=q₂−2(−1)^N−1b^(N)₂₁ ,

(32)

¯

q_1N=q¯₁+2c^(N₁₁⁾, q¯_2N=q¯₂−2c^(N)₁₂ , (33)

¯

q_1N=q¯₁+2c^(N₂₂⁾, q¯_2N=q¯₂+2c^(N)₂₁ , (34) which can be reduced to (28) by virtue of the rela- tions (A.2a) – (A.2d) in Lemma3.

As suggested by Proposition 1, the Darboux ma- trix T_N(λ) assures that the new eigenfunctionΨ_N = T_N(λ)Ψ also satisfies Lax pair (3) for the new poten- tialsq_1N andq_2N in (28). That is to say, the compat- ibility conditionΨN,tz=ΨN,zt yields the same (2) ex- cept forq_1Nandq_2N instead ofq₁andq₂, respectively.

Therefore, for a given set of linearly-independent so- lutions{Ψ_k,Ψ_k⁰,Φk,Φ_k⁰}^N_k=1of Lax pair (3), the eigen- function transformation (4) and potential transforma- tions (28) constitute theNth iterated DT:(Ψ,q₁,q₂)→

(ΨN,q_1N,q2N)of (2), where the Darboux matrixTN(λ) is determined by (17a) – (17d).

3. Soliton Solutions in Terms of the Four-Component Wronskians

In this section, we will derive the four-component Wronskian representation of the N-soliton solutions to (2) by the aboveNth iterated DT algorithm start- ing fromq1=q2=0. On this basis, we will find the parametric conditions for the generation of one- and double-hump solitons, and analyze the collisions of de- generate and non-degenerate coupled solitons in (2).

For convenience, we use the subscriptsRandIto rep- resent the real and imaginary parts, respectively.

For a given set of complex parameters {λ_k}^N_k=1 (λ_k6=λ_l), we solve Lax pair (3) withq₁=q₂=0 and λ=λk, obtaining

e_k,f_k,g_k,h_kT

= α_ke^θk,β_ke^θk,γ_ke^−θk,δ_ke^−θkT

(1≤k≤N),(35) where the phaseθ_k=−iλ_kt−2 iλ_k²z;α_k,β_k,γ_k, andδ_k are complex constants. Substitution of (35) into (28) gives the four-component Wronskian solutions to (2) as follows:

q_1N=2(−1)^Nχ11

τ , q_2N=2χ12

τ , (36) with

χ₁₁=

Λ₁Θ₊K_+,N+1 Λ₂Θ₊K_+,N Λ₃Θ−K−,N−1 Λ₄Θ−K−,N

Λ2Θ+K+,N+1 −Λ₁Θ+K+,N Λ4Θ−K−,N−1 −Λ₃Θ−K−,N

−Λ₃^∗Θ₋^∗K_−,N+1^∗ Λ₄^∗Θ₋^∗K_−,N^∗ Λ₁^∗Θ₊^∗K_+,N−1^∗ −Λ₂^∗Θ₊^∗K_+,N^∗ Λ₄^∗Θ₋^∗K_−,N+1^∗ −Λ₃^∗Θ₋^∗K_−,N^∗ −Λ₂^∗Θ₊^∗K_+,N−1^∗ −Λ₁^∗Θ₊^∗K_+,N^∗

, (37)

χ12=

Λ1Θ+K_+,N+1 Λ2Θ+K_+,N Λ3Θ−K−,N Λ4Θ−K−,N−1

Λ₂Θ+K+,N+1 −Λ₁Θ+K+,N Λ₄Θ−K−,N −Λ₃Θ−K−,N−1

−Λ₃^∗Θ₋^∗K_−,N+1^∗ Λ₄^∗Θ₋^∗K_−,N^∗ Λ₁^∗Θ₊^∗K_+,N^∗ −Λ₂^∗Θ₊^∗K_+,N−1^∗ Λ₄^∗Θ₋^∗K_−,N+1^∗ −Λ₃^∗Θ₋^∗K_−,N^∗ −Λ₂^∗Θ₊^∗K_+,N^∗ −Λ₁^∗Θ₊^∗K_+,N−1^∗

, (38)

τ=

Λ1Θ+K+,N Λ2Θ+K+,N Λ3Θ−K−,N Λ4Θ−K−,N

Λ2Θ+K_+,N −Λ₁Θ+K_+,N Λ4Θ−K−,N −Λ₃Θ−K−,N

−Λ₃^∗Θ₋^∗K_−,N^∗ Λ₄^∗Θ₋^∗K_−,N^∗ Λ₁^∗Θ₊^∗K_+,N^∗ −Λ₂^∗Θ₊^∗K_+,N^∗ Λ₄^∗Θ₋^∗K_−,N^∗ −Λ₃^∗Θ₋^∗K_−,N^∗ −Λ₂^∗Θ₊^∗K_+,N^∗ −Λ₁^∗Θ₊^∗K_+,N^∗

, (39)

whereΛ1=diag(α₁, . . .,α_N),Λ2=diag(β₁, . . .,βN), Λ3 =diag(γ₁, . . .,γ_N), Λ4 =diag(δ₁, . . .,δ_N), Θ+ = diag e^θ1, . . .,e^θN

,Θ−=diag e^−θ1, . . .,e^−θN

,K+,M=

(−iλ_n)^m−1

N×M,K−,M =

(iλ_n)^m−1

N×M (M=N− 1,N,N+1). Note from Lemma 3 that τ is a real function, and χi j and χ_{i j}⁰ (1 ≤ i,j ≤ 2) obey the

relations:

χ¯11=χ₁₁⁰ , χ¯12=χ₂₁⁰ , χ¯21=χ₁₂⁰ , χ¯22=χ₂₂⁰ . (40) On the other hand, via the Laplace expansion tech- nique, we can obtain the following four-component Wronskian identity:

τ τ_tt−τ_t²

=4(χ₁₁χ₁₁⁰ +χ₁₂χ₂₁⁰ +χ₂₁χ₁₂⁰ +χ₂₂χ₂₂⁰ ). (41) Combining (40) and (41), we have

2(|q_1N|²+|q_2N|²)

=4(χ₁₁χ¯11+χ12χ¯12+χ21χ¯21+χ22χ¯22) τ²

=τ τtt−τ_t² τ² ,

(42)

which implies that the function τhas no zeros in the tz-plane unless αk =βk =γk =δk =0 for all 1≤

4 2 0 2 4

t 1

2 3 4

q12

4 2 0 2 4

t 1

2 3 4

q12

4 2 0 2 4

t 1

2 3 4

q12

(a) (b)

= 1.15 _|μ^Ξ₁¹_ν1| ≈2.02

(c)

= 1.085928 _|μ^Ξ1

1ν₁| ≈3.00 β1I=−1.25 _|μ^Ξ₁¹_ν1| ≈1.43 β_1I β1I

Fig. 1. Non-degenerate one-hump solitons forq₁via (45) transverse atz=0.45, with the parameters chosen asβ1=−0.05+ β1Ii,γ1=0.01+0.01 i,δ1=0.01−0.02 i, andλ1=−1+i.

6 3 0 3 6

t 1

2

q12

6 3 0 3 6

t 1

2

q12

5 5 15 25 35

t 1

2

q12

(a)

β_1R= 0.2 _|μ^Ξ1

1ν₁| ≈11.26

(b)

β_1R= 0.004 _|μ^Ξ1

1ν₁| ≈559.02 (c)

β_1R= 4×10⁻²⁷ _|μ^Ξ1

1ν₁| ≈5.59×10²⁶ Fig. 2. Non-degenerate double-hump solitons forq1via (45) transverse atz=0, with the parameters chosen asβ1=β1R+i, γ1=−0.1−0.1 i,δ1=−0.1, andλ1=−1+i. The distance between two humps are respectively as follows: (a)d1≈1.47;

(b)d1≈3.51; (c)d1≈31.14.

k≤N. Therefore, we can safely say that (36) together with (37) – (39) represent theN-soliton solutions to (2) ifα_k,β_k,γ_k, andδ_k are not all equal to zero. As re- marked in [11–13], we can without loss of general- ity takeα_k=1 for 1≤k≤N. That is to say, theN- soliton solutions to (2) are in general characterized by 4N complex parameters {β_k,γk,δk,λ_k}^N_k=1, which is greater than the number of those obtained by the non- standard Hirota method in [24,25].

WithN=1, (36) imply the following three families of one-soliton solutions:

q₁ q₂

=λ1Iµ1e^θ1−θ¯1

|µ₁κ1| sech

ξ1+ln |µ₁| 2|κ₁|

κ¯1

κ¯₂

(γ₁=±iδ₁), (43) q₁

q2

=λ_1Iν¯₁e^θ1−θ¯1

|ν₁κ1| sech

ξ₁+ln2|κ₁|

|ν₁| κ₁

κ2

(β₁=±i), (44)

q₁ q₂

= λ1Ie^θ1−θ¯1

|µ₁ν1|cosh² ξ1+ln q_|µ

1|

|ν₁|

+Ξ1− |µ₁ν1|

·

"

e^−ξ1 ν¯1κ1−^|µ1_¯^ν1|

µ1 κ¯1

ν¯1κ2−^|µ1_¯^ν1|

µ₁ κ¯2

!

+2µ1

s|ν₁|

|µ₁|cosh ξ1+ln s|µ₁|

|ν₁|

!

· κ¯1

κ¯₂

(β₁6=±i,γ16=±iδ₁),

(45)

with

µ1=1+β₁², ν1=γ₁²+δ₁², κ1=γ1+β1δ1, κ2=δ1−β1γ1, ξ1=θ1+θ¯1, Ξ1= β1−β¯1

· δ₁γ¯₁−γ₁δ¯₁

+ (1+|β₁|²)(|γ₁|²+|δ₁|²).

Solutions (43) and (44) are both degenerate in the sense that they represent only the one-hump solitons. In both the two cases, the componentsq1andq2have the same intensities. Solution (45) is non-degenerate because it could describe either the one- or double-hump soliton, depending on the choice of parameters.

In order to figure out the dependence of the soliton profiles on the parameters in (45), we take the deriva- tives of|q_j|²(j=1,2) with respect toξ1, giving that

∂|q_j|²

∂ ξ1

= 32λ_1I²e^2ξ1 |µ₁|²e^4ξ1− |ν₁|²

|µ₁|²e^4ξ1+2Ξ1e^2ξ1+|ν₁|²3

·

|κ_j|²|µ₁|²e^4ξ1+|κ_j|²|ν₁|²+2 µ1ν1κ¯²_j +µ¯1ν¯1κ²_j− |κ_j|²Ξ1

e^2ξ1

(j=1,2),

(46)

which suggests that the maxima of|q_j|²are related to

∆j(j=1,2) defined by

∆1=κ₁² δ¯₁²+β¯₁²γ¯₁²

+κ¯₁² δ₁²+β₁²γ₁² + β1γ¯1δ1−β¯1γ1δ¯1

2

+ |γ₁|²− |β₁|²|δ₁|²2

,

(47)

∆2=κ₂² γ¯₁²+β¯₁²δ¯₁²

+κ¯₂² γ₁²+β₁²δ₁² + β₁δ¯₁γ₁−β¯₁δ₁γ¯₁2

+ |δ₁|²− |β₁|²|γ₁|²2

.

(48) For the case∆j≤0, it is seen from (46) that|q_j|²has only one maximum

|q_j|²_max=4λ_1I²|ν₁|

|µ1|·

µ1|ν₁|κ¯j+|µ₁|ν¯1κj

2

(|ν₁|Ξ1+|µ₁||ν₁|²)² (j=1,2)

(49)

along the line ξ1 = ¹₂ln^|ν_|µ^1|

1| in the tz-plane. Fig- ures 1a – c present three types of one-hump solitons with the parameters satisfying ∆j ≤0. One can ob- serve that the top of|q_j|²tends to be flatter with the increase of_|µ^Ξ1

1ν1|. If∆_j>0, the non-degenerate soliton exhibiting the double-hump profile is characterized by the following three features: (i) The two humps in|q_j|² are symmetric and have the same height, that is,

|q_j|²_max=

16λ_1I²|κ_j|⁴

p∆_j+|κ_j|²Ξ₁−µ₁ν₁κ¯²_j

2hp

∆_j−2 Re(µ¯₁ν¯₁κ²_j) +|κ_j|²Ξ₁ i ∆_j+4p

∆_j|κ_j|²Ξ₁+3|κ_j|⁴(Ξ₁²+|µ₁|²|ν₁|²) +2 Re(Ω_j)2 (j=1,2), (50) with Ωj = µ₁²ν₁²κ¯⁴_j − 4µ1ν1Ξ1|κ_j|²κ¯²_j − 2µ1ν1

p∆_jκ¯²_j, which is reached along two lines in the tz-plane:

ξ1= 1

2ln|κ_j|²Ξ₁−µ₁ν₁κ¯²_j−µ¯₁ν¯₁κ²_j±p

∆_j

|µ₁|²|κ_j|²

(j=1,2). (51)

(ii) The soliton does not change its shape and remains the separation between the two humps during propa- gation. (iii) The formulae for the distance between the

two humps in|q_j|²is explicitly given by

d_j= 1 4λ1I

ln|κ_j|²Ξ1−µ1ν1κ¯²_j−µ¯1ν¯1κ²_j+p

∆_j

|κ_j|²Ξ1−µ1ν1κ¯²_j−µ¯1ν¯1κ²_j−p

∆j

(52) (j=1,2), which tells us that one hump will be separated with the other one further and even to the infinity as the value of _|µ^Ξ1

1ν1|increases, as displayed in Figures2a – c.

ForN≥2, (36) can describe the dynamics of cou- pled soliton collisions in (2). Here, we make the asymptotic analysis of (36) withN=3, finding the fol- lowing collision properties: (i) In the collisions among

16 8 0 8 16 t

1 2 3 4

q12

S11

S12

S13

S11 S12

S13

16 8 0 8 16

t 1

2 3 4

q22

S21

S22

S23

S21 S22

S23

(a) (b)

Fig. 3 (colour online). Shape-preserving collisions among three degenerate solitons via (45) transverse respectively atz=−15 (for the left solitons) andz=15 (for the right solitons), withN=3,β1=i,β2=2−2 i,β3=i,γ1=2,γ2=2 i,γ3=3−i, δ1=1−i,δ2=2,δ3=1,λ1=−0.1+1.2 i,λ2=−0.15−i, andλ3=−0.2−1.5 i.

16 8 0 8 16

t 1

2 3 4

q12

S11

S12

S13

S11

S12

S13

16 8 0 8 16

t 2

4 6 8

q22

S₂₁ S22

S23

S₂₁ S22

S23

(a) (b)

Fig. 4 (colour online). Shape-preserving collisions among three non-degenerate solitons via (45) transverse respectively at z=−15 (for the left solitons) andz=15 (for the right solitons), withN=3,β1=1+2 i,β2=2,β3=2,γ1=2 i,γ2=1+2 i, γ3=−i,δ1=1,δ2=1,δ3=0,λ1=−0.1+1.2 i,λ2=−0.15−i, andλ3=−0.2−1.5 i.

20 10 0 10 20

t 1

2 3 4

q12

S11

S₁₂ S13

S11S12

S13

20 10 0 10 20

t 1

2 3 4

q22

S21

S22

S23

S21S22

S₂₃

(a) (b)

Fig. 5 (colour online). Shape-changing collisions of one degenerate soliton with two non-degenerate solitons via (45) trans- verse respectively atz=−20 (for the left solitons) andz=20 (for the right solitons), where the degenerate soliton(S₁₁,S₂₁) experiences no change upon collision, while the non-degenerate solitons(S12,S22)and(S13,S23)change their amplitudes and profiles after collision. The parameters are chosen asN=3,β1=i,β2=1,β3=2,γ1=2,γ2=2+i,γ3=−i,δ1=1, δ2=1,δ3=0.8,λ1=−0.1+1.2 i,λ2=−0.15−i, andλ3=−0.2−1.5 i.

degenerate (or non-degenerate) solitons, both|q₁|²and

|q₂|² meet the conventional elastic case, that is, each component of interacting solitons maintain individ- ual shape, velocity, and energy after collision. (ii) In the case of degenerate solitons interacting with non- degenerate ones, the degenerate ones in both two com- ponents experience always the elastic collisions, while

the non-degenerate ones may undergo the change of shapes including the soliton profiles and amplitudes, together with the energy redistribution between two components. The above two properties coincide with those obtained in [24,25]. Some examples of soliton collisions via (36) withN =3 are illustrated in Fig- ures3–5, whereS⁻_jnandS⁺_jn(n=1,2,3) represent the