Conservation Laws and Mixed-Type Vector Solitons for the 3-Coupled Variable-Coefficient Nonlinear Schrödinger Equations in Inhomogeneous Multicomponent Optical Fibre

Jun Chai; Bo Tian; Yu-Feng Wang; Wen-Rong Sun; Yun-Po Wang

doi:10.1515/zna-2016-0019

Artikel Öffentlich zugänglich

Conservation Laws and Mixed-Type Vector Solitons for the 3-Coupled Variable-Coefficient Nonlinear Schrödinger Equations in Inhomogeneous Multicomponent Optical Fibre

Jun Chai , Bo Tian , Yu-Feng Wang , Wen-Rong Sun und Yun-Po Wang

Veröffentlicht/Copyright: 22. April 2016

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Zeitschrift für Naturforschung A Band 71 Heft 6

Abstract

In this article, the propagation and collision of vector solitons are investigated from the 3-coupled variable-coefficient nonlinear Schrödinger equations, which describe the amplification or attenuation of the picosecond pulses in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations. On the basis of the Lax pair, infinitely-many conservation laws are obtained. Under an integrability constraint among the variable coefficients for the group velocity dispersion (GVD), nonlinearity and fibre gain/loss, and two mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation. Influence of the variable coefficients for the GVD and nonlinearity on the vector soliton amplitudes and velocities is analysed. Through the asymptotic and graphic analysis, bound states and elastic and inelastic collisions between the vector two solitons are investigated: Not only the elastic but also inelastic collision between the 2-bright-1-dark vector two solitons can occur, whereas the collision between the 1-bright-2-dark vector two solitons is always elastic; for the bound states, the GVD and nonlinearity affect their types; with the GVD and nonlinearity being the constants, collision period decreases as the GVD increases but is independent of the nonlinearity.

Keywords: 3-Coupled Variable-Coefficient Nonlinear Schrödinger Equations; Infinitely-Many Conservation Laws; Mixed-Type Vector Soliton; Multicomponent Optical Fibre

1 Introduction

Optical solitons, formed via the balance between the group velocity dispersion (GVD) and nonlinearity [1–4], have been the subject of theoretical and experimental studies in the optical-fibre communications, due to the capability of propagating over the long distances without any change of shape and with the negligible attenuation [5–8]. Propagation of the optical solitons in the single-component fibres can be described by some nonlinear Schrödiger (NLS)-type equations [9, 10]. With the consideration on the collisions of those field components with the different frequencies or polarizations in such optical media as the multimode fibres, fibre arrays and birefringent fibres, the coupled NLS equations (e.g. the two-, three-, and N-coupled ones) have been used to describe the propagation of the vector solitons [11]. Compared with the scalar soliton which has only one polarization component, a vector soliton has the multiple distinct polarization components coupled together during the propagation [12, 13]. One phenomenon associated with the collisions of the vector solitons is the energy exchange among the components [14]. So far, three types of the vector solitons have been proposed including the bright vector solitons (all of the polarization states are the bright solitons), dark vector solitons (all dark), and mixed-type vector solitons (some bright while the others dark) [15–17]. Propagation of the vector solitons can be described by a system of the coupled NLS equations [9, 18].

Investigation on the coupled NLS equations with constant coefficients as the models of some physical problems has been performed [19–22]. Propagation of optical solitons in the inhomogeneous optical fibres can be modelled by the variable-coefficient NLS equations [5, 23–26]. Moreover, some studies on the coupled variable-coefficient NLS equations have been presented [5, 26–32]. In this article, we investigate the propagation and collision of vector optical solitons from the 3-coupled variable-coefficient NLS equations [5],

(1a)iq1,z + 12β(z)q1,tt + γ(z)(∑n = 13|qn|2)q1 + iδ(z)q1 = 0, (1a)

(1b)iq2,z + 12β(z)q2,tt + γ(z)(∑n = 13|qn|2)q2 + iδ(z)q2 = 0, (1b)

(1c)iq3,z + 12β(z)q3,tt + γ(z)(∑n = 13|qn|2)q3 + iδ(z)q3 = 0, (1c)

which have been considered as a model to describe the amplification or attenuation of the picosecond pulse propagation in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations, where q_j is the complex amplitude of the j-th-field component (j=1, 2, 3), the subscripts z and t, respectively, represent the partial derivatives with respect to the normalised distance along the direction of the propagation and retarded time, β(z), γ(z), and δ(z) are the GVD, nonlinearity, and fibre gain/loss coefficients, respectively [5]. Hereby, the effect of the GVD, which can broaden a soliton, can be counterbalanced by the action of nonlinearity [11]. Therefore, it has been claimed that the soliton can use the nonlinearity to maintain its width in the presence of the GVD [11]. However, the property holds only if the fibre gain/loss is negligible, as the fibre gain/loss can lead to the change of the soliton energy [11]. Lax integrability for (1) has been confirmed under the constraint [5]

(2)δ(z) = γ′(z)β(z) − γ(z)β′(z)2γ(z)β(z) or δ(z) = [lnγ(z)β(z)]′, (2)

where “′” means the derivative with respect to z. Darboux transformation has also been derived [5].

However, characteristics for (1), such as the infinitely-many conservation laws, bilinear forms, and mixed-type vector soliton solutions, have not yet been investigated. In Section 2, the infinitely-many conservation laws will be obtained. In Section 3, with the Hirota method [33, 34] and symbolic computation [35–37], bilinear forms and analytic mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions for (1) will be given. Influence of the variable coefficients (GVD and nonlinearity) in (1) on the vector soliton amplitudes, velocities, and the vector soliton collisions will be analysed in Section 4. Section 5 will be our conclusions.

2 Conservation Laws

Conservation laws, which describe the conservation of physical quantities for a nonlinear evolution equation (NLEE) [38, 39], are the integrable characteristic for the NLEE. Hereby, on the basis of the Lax pair given in [5], we will construct the infinitely-many conservation laws for (1).

The corresponding Lax pair can be expressed as [5]

(3)Ψt = UΨ, Ψz = VΨ, (3)

where the vector eigenfunction Ψ=(Ψ₀, Ψ₁, Ψ₂, Ψ₃)^T, T denotes the transpose of the vector, the components Ψ₀, Ψ₁, Ψ₂, and Ψ₃ are the complex functions of z and t, and U and V are expressible in the forms of

U = (−iλγ(z)2β(z)q1γ(z)2β(z)q2γ(z)2β(z)q3−γ(z)2β(z)q1∗iλ00−γ(z)2β(z)q2∗0iλ0−γ(z)2β(z)q3∗00iλ),V = (−2iλ2γ(z)2β(z)q1λγ(z)2β(z)q2λγ(z)2β(z)q3λ−γ(z)2β(z)q1∗λ2iλ200−γ(z)2β(z)q2∗λ02iλ20−γ(z)2β(z)q3∗λ002iλ2)+ i2(γ(z)∑j = 13|qj|22γ(z)β(z)q1t2γ(z)β(z)q2t2γ(z)β(z)q3t2γ(z)β(z)(q1∗)t−γ(z)|q1|2−γ(z)q2q1∗−γ(z)q3q1∗2γ(z)β(z)(q2∗)t−γ(z)q1q2∗−γ(z)|q2|2−γ(z)q3q2∗2γ(z)β(z)(q3∗)t−γ(z)q1q3∗−γ(z)q2q3∗−γ(z)|q3|2),

with λ as the spectral parameter.

Introducing the functions Γj = ΨjΨ0 (j = 1, 2, 3), we get the following Ricatti equations:

(4)Γj,t = − ϱqj∗ + 2iλΓj − ∑l = 13(ϱqlΓl)Γj, (4)

with ϱ = γ(z)2β(z).

We expand ϱq_jΓ_j as a power series of λ⁻¹, namely,

(5)ϱqjΓj = ∑n = 1∞Ϝn(j)λ−n, (5)

where Ϝn(j)’s (n=1, 2, …) are the functions of z and t to be determined.

Then, substituting Expression (5) into Expression (4) and equating the coefficients of the same power of λ to zero, we can obtain the recurrence relations,

(6a)Ϝ1(j) = 12iϱ2|qj|2, (6a)

(6b)Ϝ2(j) = 12i{−qjtqjϜ1(j) + Ϝ1,t(j)}, (6b)

(6c)Ϝr + 1(j) = 12i{−qjtqjϜr(j) + ∑l = 1r − 1[∑s = 13Ϝl(s)]Ϝr − l(j) + Ϝr,t(j)}, (r = 2, 3, 4 …). (6c)

Through the compatibility condition (ln Ψ₀)_tz=(ln Ψ₀)_zt, we can obtain the following conservation form:

(7)[−iλ + ϱq1Γ1 + ϱq2Γ2 + ϱq3Γ3]z = [Q0 + Q1Γ1 + Q2Γ2 + Q3Γ3]t, (7)

with

Q0 = − 2iλ2 + i2γ(z)∑j = 13|qj|2,Q1 = ϱq1λ + i22γ(z)β(z)q1t,Q2 = ϱq2λ + i22γ(z)β(z)q2t,Q3 = ϱq3λ + i22γ(z)β(z)q3t.

Substituting Expressions (5) and (6) into Expression (7) yields the infinitely-many conservation laws,

(8)∂∂zRn + ∂∂tJn = 0, n = 1, 2, …, (8)

with

(9)R1 = − 12iϱ2∑j = 13|qj|2, (9)

(10)J1 = 14ϱ2∑j = 13[qj(qj∗)t] − 14γ(z)∑j = 13(qj∗qjt), (10)

(11)R2 = − 14ϱ2∑j = 13[qj(qj∗)t], (11)

(12)J2 = − i8ϱ4∑j = 13|qj|4 − i4ϱ4∑j,l = 1,j ≠ l3(|qj|2|ql|2) + i8γ(z)∑j = 13[qjt(qj∗)t]−i8ϱ2∑j = 13[qj(qj∗)tt], (12)

(13)R3 = 18iϱ2{ϱ2∑j = 13|qj|4 + 2ϱ2∑j,l = 1,j ≠ l3(|qj|2|ql|2) + ∑j = 13[qj(qj∗)tt]}, (13)

(14)J3 = 116γ(z)ϱ2∑j,l = 13(|ql|2qj∗qjt) + 116γ(z)∑j = 13[qjt(qj∗)tt], (14)

⁝

where R_n’s and J_n’s represent the conserved densities and conserved fluxes, respectively.

3 Bilinear Forms and Vector Soliton Solutions for (1)

3.1 Bilinear Forms

Under Constraint (2), introducing the following dependent variable transformations [23]:

(15)qj = β(z)γ(z)g(j)f, (j = 1, 2, 3), (15)

where g^(j)’s are all the complex functions to be determined and f is a real one, we can transform (1) into the bilinear forms

(16a)[iDz + β(z)2Dt2 − ζ(z)]g(j) ⋅ f = 0, (16a)

(16b)[β(z)2Dt2 − ζ(z)]f ⋅ f = β(z)∑j = 13|g(j)|2, (16b)

where ζ(z) is a real function to be determined, and the bilinear operators D_z and D_t are defined by [33, 34]:

Dzm1Dtm2(G ⋅ F) = (∂∂z − ∂∂z′)m1(∂∂t − ∂∂t′)m2G(z, t)F(z′, t′)|z′ = z,t′ = t,

with G(z, t) as a differentiable function of z and t, F(z′, t′) as a differentiable function of the formal variables z′ and t′, m₁, and m₂ as two non-negative integers.

3.2 Vector Soliton Solutions

3.2.1 2-Bright-1-Dark Vector Soliton Solutions

In order to obtain the 2-bright-1-dark vector soliton solutions for (1), we expand g^(j) (j=1, 2, 3) and f with respect to a formal expansion parameter ε as follows:

(17a)g(j) = εg1(j) + ε3g3(j) + ε5g5(j) + ⋯, (j = 1, 2), (17a)

(17b)g(3) = g0(3)[1 + ε2g2(3) + ε4g4(3) + ε6g6(3) + ⋯], (17b)

(17c)f = 1 + ε2f2 + ε4f4 + ε6f6 + ⋯, (17c)

where gO(j)’s (O=1, 3, 5, …) and gL(3)’s (L=0, 2, 4, 6, …) are all the complex functions to be determined, f_W’s (W=2, 4, 6, …) are the real functions to be determined.

(A) 2-bright-1-dark vector one-soliton solutions

Truncating Expression (17) as g(j) = εg1(j) (j = 1, 2),g(3) = g0(3)[1 + ε2g2(3)] and f=1+ε²f₂, and substituting them into Bilinear Forms (16), we obtain the 2-bright-1-dark vector one-soliton solutions for (1) as

(18a)qj = β(z)γ(z)g1(j)1 + f2, (j = 1, 2), (18a)

(18b)q3 = β(z)γ(z)g0(3)[1 + g2(3)]1 + f2, (18b)

where

g1(j) = a(j)eθ, g0(3) = χeϕ, g2(3) = κeθ + θ∗, f2 = τeθ + θ∗, ζ(z) = − χχ∗β(z),ϕ = ipt + i2∫[(2χχ∗ − p2)β(z)]dz, θ = ηt + i2∫[(2χχ∗ + η2)β(z)]dz,κ = (p + iη)τp − iη∗, τ = [a(1) + a(1)∗ + a(2) + a(2)∗](p + iη)(p − iη∗)(η + η∗)2[χχ∗ + (p + iη)(p − iη∗)],

with * as the complex conjugate, a^(j)’s, χ, η, and p as all the constants.

(B) 2-bright-1-dark vector two-soliton solutions

To derive the 2-bright-1-dark vector two-soliton solutions for (1), we truncate Expression (17) as g(j) = εg1(j) + ε3g1(j) (j = 1, 2),g(3) = g0(3)[1 + ε2g2(3) + ε4g4(3)] and f=1+ε²f₂+ε⁴f₄, and substitute them into Bilinear Forms (16). Then we get

(19a)qj = β(z)γ(z)g1(j) + g3(j)1+f2 + f4, (j = 1, 2), (19a)

(19b)q3 = β(z)γ(z)g0(3)[1 + g2(3) + g4(3)]1 + f2 + f4, (19b)

where

g1(j) = a1(j)eθ1 + a2(j)eθ2, g3(j) = A1(j)eθ1 + θ2 + θ1∗ + A2(j)eθ1 + θ2 + θ2∗, g0(3) = χeϕ,g2(3) = κ1eθ1 + θ1∗ + κ2eθ2 + θ1∗ + κ3eθ1 + θ2∗ + κ4eθ2 + θ2∗, g4(3) = ρeθ1 + θ1∗ + θ2 + θ2∗,f2 = τ1eθ1 + θ1∗ + τ2eθ2 + θ1∗ + τ3eθ1 + θ2∗ + τ4eθ2 + θ2∗, f4 = heθ1 + θ1∗ + θ2 + θ2∗,θ1 = η1t + i2∫[(2χχ∗ + η12)β(z)]dz, θ2 = η2t + i2∫[(2χχ∗ + η22)β(z)]dz,ϕ = ipt + i2∫[(2χχ∗ − p2)β(z)]dz, ζ(z) = − χχ∗β(z), A1(j) = (η1 − η2)[a1(j)τ2η1 + η1∗ − a2(j)τ1η2 + η1∗], A2(j) = (η1 − η2)[a1(j)τ4η1 + η2∗ − a2(j)τ3η2 + η2∗],κ1 = (p + iη1)τ1p − iη1∗, κ2 = (p + iη2)τ2p − iη1∗, κ3 = (p + iη1)τ3p − iη2∗, κ4 = (p + iη2)τ4p − iη2∗,τ1 = [a1(1)a1(1)∗ + a1(2)a1(2)∗](p + iη1)(p − iη1∗)(η1 + η1∗)2[(p + iη1)(p − iη1∗) + χχ∗],τ2 = [a2(1)a1(1)∗ + a2(2)a1(2)∗](p + iη2)(p − iη1∗)(η2 + η1∗)2[(p + iη2)(p − iη1∗) + χχ∗],τ3=[a1(1)a2(1)∗ + a1(2)a1(2)∗](p + iη1)(p − iη2∗)(η1 + η2∗)2[(p + iη1)(p − iη2∗) + χχ∗],τ4 = [a2(1)a2(1)∗ + a2(2)a2(2)∗](p + iη2)(p − iη2∗)(η2 + η2∗)2[(p + iη2)(p − iη2∗) + χχ∗],ρ = h(p + iη1)(p + iη2)(p − iη1∗)(p − iη2∗), h = τ1τ4(η1 − η2)(η1∗ − η2∗)(η1 + η2∗)(η2 + η1∗) − τ2τ3(η1 − η2)(η1∗ − η2∗)(η1 + η1∗)(η2 + η2∗),

with a1(j)’s,a2(j)’s,χ, η₁, η₂, and p as all the constants.

3.2.2 1-Bright-2-Dark Vector Soliton Solutions

Similarly, we take

(20a)g(1) = εg1(1) + ε3g3(1) + ε5g5(1) + ⋯ , (20a)

(20b)g(j) = g0(j)[1 + ε2g2(j) + ε4g4(j) + ε6g6(j) + ⋯], (j = 2, 3) (20b)

(20c)f = 1 + ε2f2 + ε4f4 + ε6f6 + ⋯, (20c)

where gl(1)’s (l=1, 3, 5, …) and gb(j)’s (b=0, 2, 4, 6, …) are all the complex functions to be determined, f_w’s (w=2, 4, 6, …) are the real functions to be determined.

(A) 1-bright-2-dark vector one-soliton solutions

We truncate Expression (20) as g(1) = εg1(1),g(j) = g0(j)[1 + ε2g2(j)] (j = 2, 3),f=1+ε²f₂, substitute them into Bilinear Forms (16), and obtain

(21a)q1 = β(z)γ(z)g1(1)1 + f2, (21a)

(21b)qj = β(z)γ(z)g0(j)[1 + g2(j)]1 + f2, (j = 2, 3), (21b)

where

g(1) = aeΘ, g0(j) = χjeϕj, g2(j) = τjeΘ + Θ∗, f2 = κeΘ + Θ∗,ϕj = ipjt + i∫[(− 12pj2 + χ2χ2∗ + χ3χ3∗)β(z)]dz,Θ = μt + i∫[(− 12μ2 + χ2χ2∗ + χ3χ3∗)β(z)]dz,λ(z) = − χ2χ2∗β(z) − χ3χ3∗β(z), τj = κ(pj + iμ)(pj − iμ∗),κ = aa∗H(μ + μ∗)2[H + (p3 + iμ)(p3 − iμ∗)χ2χ2∗ + (p2 + iμ)(p2 − iμ∗)χ3χ3∗],H = (p2 + iμ)(p2 − iμ∗)(p3 + iμ)(p3 − iμ∗),

with a, χ_j’s, μ, and p_j’s as all the constants.

(B) 1-bright-2-dark vector two-soliton solutions

With the truncations of Expression (20) g(1) = εg1(1) + ε3g3(1),g(j) = g0(j)[1 + ε2g2(j) + ε4g4(j)] (j = 2, 3) and f=1+ε²f₂+ε⁴f₄, we get the vector two-soliton solutions for (1),

(22a)q1 = β(z)γ(z)g1(1) + g3(1)1 + f2 + f4, (22a)

(22b)qj = β(z)γ(z)g0(j)[1 + g2(j) + g4(j)]1 + f2 + f4, (j = 2, 3), (22b)

where

g1(1) = a1eΘ1 + a2eΘ2, g3(1) = A1eΘ1 + Θ2 + Θ1∗ + A2eΘ1 + Θ2 + Θ2∗, g0(j) = χjeϕj,g2(j) = κ1(j)eΘ1 + Θ1∗ + κ2(j)eΘ2 + Θ1∗ + κ3(j)eΘ1 + Θ2∗ + κ4(j)eΘ2 + Θ2∗, g4(j) = ς(j)eΘ1 + Θ2 + Θ1∗ + Θ2∗, f2 = σ1eΘ1 + Θ1∗ + σ2eΘ2 + Θ1∗ + σ3eΘ1 + Θ2∗ + σ4eΘ2 + Θ2∗, f4 = νeΘ1 + Θ2 + Θ1∗ + Θ2∗, λ(z) = − χ2χ2∗β(z) − χ3χ3∗β(z),ϕj = ipjt + i∫[(− 12pj2 + χ2χ2∗ + χ3χ3∗)β(z)]dz,Θ1 = μ1t + i∫[(− 12μ12 + χ2χ2∗ + χ3χ3∗)β(z)]dz,Θ2 = μ2t + i∫[(− 12μ22 + χ2χ2∗ + χ3χ3∗)β(z)]dz,κ1(j) = (pj + iμ1)σ1pj − iμ1∗, κ2(j) = (pj + iμ2)σ2pj − iμ1∗,κ3(j) = (pj + iμ1)σ3pj − iμ2∗, κ4(j) = (pj + iμ2)σ4pj − iμ2∗,σ1 = a1a1∗H1(μ1 + μ1∗)2[H1 + (p3 + iμ1)(p3 − iμ1∗)χ2χ2∗ + (p2 + iμ1)(p2 − iμ1∗)χ3χ3∗],σ2 = a2a1∗H2(μ2 + μ1∗)2[H2 + (p3 + iμ2)(p3 − iμ1∗)χ2χ2∗ + (p2 + iμ2)(p2 − iμ1∗)χ3χ3∗],σ3 = a1a2∗H3(μ1 + μ2∗)2[H3 + (p3 + iμ1)(p3 − iμ2∗)χ2χ2∗ + (p2 + iμ1)(p2 − iμ2∗)χ3χ3∗],σ4 = a2a2∗H4(μ2 + μ2∗)2[H4 + (p3 + iμ2)(p3 − iμ2∗)χ2χ2∗ + (p2 + iμ2)(p2 − iμ2∗)χ3χ3∗],H1 = (p2 + iμ1)(p2 − iμ1∗)(p3 + iμ1)(p3 − iμ1∗),H2 = (p2 + iμ2)(p2 − iμ1∗)(p3 + iμ2)(p3 − iμ1∗),H3 = (p2 + iμ1)(p2 − iμ2∗)(p3 + iμ1)(p3 − iμ2∗),H4 = (p2 + iμ2)(p2 − iμ2∗)(p3 + iμ2)(p3 − iμ2∗),A1 = − (η1 − η2)[a2σ1(η1 + η1∗) − a1σ3(η1∗ + η2)](η1 + η1∗)(η2 + η1∗),A2 = − (η1 − η2)[a2σ2(η1 + η2∗) − a1σ4(η2∗ + η2)](η1 + η2∗)(η2 + η2∗),ς(j) = ν(pj + iμ1)(pj + iμ2)(pj − iμ1∗)(pj − iμ2∗), ν = σ1σ4(μ1 − μ2)(μ1∗ − μ2∗)(μ1 + μ2∗)(μ2 + μ1∗) − σ2σ3(μ1 − μ2)(μ1∗ − μ2∗)(μ1+μ1∗)(μ2+μ2∗),

with a₁, a₂, χ_j’s, μ₁, μ₂, and p_j’s as all the constants.

4 Discussion on the Vector Solitons

4.1 Vector One Soliton

In this section, we investigate the properties of the vector solitons based on Solutions (18) and (21), such as the velocities and amplitudes of the bright and dark solitons.

From Solutions (18), we can obtain the velocities [v_j (j=1, 2, 3)] and amplitudes [Δ_j (j=1, 2, 3)] of the 2-bright-1-dark vector soliton as

(23a)vj = Im(η)β(z) (j = 1, 2, 3), (23a)

(23b)Δj = 12|a(j)β(z)γ(z)τ| (j = 1, 2), (23b)

(23c)Δ3 = |χβ(z)γ(z)|, (23c)

with Re and Im as the real part and imaginary part of the parameter, respectively.

The velocities [v˜j(j = 1, 2, 3)] and amplitudes [Δ˜j (j = 1, 2, 3)] of the 1-bright-2-dark vector soliton via Solutions (21) are derived as

(24a)v˜j = Im(μ)β(z) (j = 1, 2, 3), (24a)

(24b)Δ˜1 = 12|aβ(z)γ(z)κ|, (24b)

(24c)Δ˜j = |χjβ(z)γ(z)| (j = 2, 3). (24c)

From Expressions (23) and (24), we find that the amplitudes of the two mixed-type vector solitons are both related to β(z) and γ(z), but the velocities of them are only related to β(z).

2-Bright-1-dark vector soliton via Solutions (18) is shown in Figures 1–3, while 1-bright-2-dark vector soliton via Solutions (21) is displayed in Figures 4–6. In Figures 1, 2, 4, and 5, with β(z)=γ(z) leading to δ(t)=0, we find that amplitudes of the vector solitons keep unchanged during the propagation. It indicates that, with the absence of the fibre gain/loss, the vector soliton can propagate without the energy change. Besides, if we consider the homogeneous optical fibre, where β(z) and γ(z) both being the constants, vector solitons propagate stably with the invariable velocities, as shown in Figures 1 and 4. However, in the inhomogeneous optical fibre, choosing β(z)=γ(z)=0.1z² in Figure 2 while β(z)=γ(z)=cos(z) in Figure 5, we find that the vector soliton propagates S-shaped and periodically, respectively. The velocity of the vector soliton shown in Figure 2 slowly decreases and eventually approaches a steady positive velocity for t→+∞ or –∞, while that shown in Figure 5 changes periodically. Amplitude-changing vector soliton can be obtained, when the value of β(z)γ(z) is variable, i.e. δ(t)≠0, as shown in Figures 3 and 6. The presence of the fibre gain/loss can cause the change of the soliton energy. In addition, owing to the different choices of β(z), the velocity of the vector soliton shown in Figure 3 gradually increases with t, while that of the vector soliton shown in Figure 6 keeps unchanged.

Figure 1:

(a–c) Three-dimensional graphs of the 2-bright-1-dark vector one soliton via Solutions (18) with a⁽¹⁾=3, a⁽²⁾=4, η=2+i, χ=1+0.5i, p=1, β(z)=γ(z)=0.5. (d–f) Contour plots of (a–c), respectively.

Figure 2:

The same as Figure 1 except that β(z)=γ(z)=0.1z².

Figure 3:

The same as Figure 1 except that β(z)=sech(0.4z), γ(z)=1.

Figure 4:

(a–c) Three-dimensional graphs of the 1-bright-2dark vector one soliton via Solutions (21) with a=3, χ₂=2, χ₃=2.2, μ=0.8+0.7i, p₂=p₃=2, β(z)=γ(z)=2. (d–f) Contour plots of (a–c), respectively.

Figure 5:

The same as Figure 4 except that β(z)=γ(z)=cos(z).

Figure 6:

The same as Figure 4 except that β(z)=2, γ(z)=2.5e^−0.15z.

4.2 Vector Two Solitons

4.2.1 Elastic and Inelastic Collisions of the Vector Two Solitons

On the basis of Solutions (19) and (22), in this part, we will investigate the elastic and inelastic collisions of the vector solitons via the asymptotic and graphical analysis, in the case that the choices of the GVD coefficient β(t) and nonlinearity coefficient γ(t) lead to the fibre gain/loss coefficient δ(t)=0.

(A) Asymptotic analysis on Solutions (19)

Before the collision (z→–∞),

(25a)qj1− → β(z)γ(z)a1(j)eθ11 + τ1eθ1 + θ1∗ (j = 1, 2), (θ1 + θ1∗ ∼ 0, θ2 + θ2∗ → − ∞), (25a)

(25b)q31− → β(z)γ(z)χeϕ(1 + κ1eθ1 + θ1∗)1 + τ1eθ1 + θ1∗, (θ1 + θ1∗ ∼ 0, θ2 + θ2∗ → − ∞), (25b)

(25c)qj2− → β(z)γ(z)A1(j)eθ2τ1 + heθ2 + θ2∗ (j = 1, 2), (θ2 + θ2∗ ∼ 0, θ1 + θ1∗ → + ∞), (25c)

(25d)q32− → β(z)γ(z)χeϕ(κ1 + ρeθ2 + θ2∗)τ1 + heθ2 + θ2∗, (θ2 + θ2∗ ∼ 0, θ1 + θ1∗ → + ∞), (25d)

with qj1−’s and qj2−’s (j=1, 2, 3) as the asymptotic expressions for the vector two solitons before the collision, respectively.

After the collision (z→+∞),

(26a)qj1+ → β(z)γ(z)A2(j)eθ1τ4 + heθ1 + θ1∗ (j = 1, 2), (θ1 + θ1∗ ∼ 0, θ2 + θ2∗ → + ∞), (26a)

(26b)q31+ → β(z)γ(z)χeϕ(τ4 + ρeθ1 + θ1∗)τ4 + heθ1 + θ1∗, (θ1 + θ1∗ ∼ 0, θ2 + θ2∗ → + ∞), (26b)

(26c)qj2+ → β(z)γ(z)a2(j)eθ21 + τ4eθ2 + θ2∗ (j = 1, 2), (θ2 + θ2∗ ∼ 0, θ1 + θ1∗ → − ∞), (26c)

(26d)q32+ → β(z)γ(z)χeϕ(1 + κ4eθ2 + θ2∗)1 + τ4eθ2 + θ2∗, (θ2 + θ2∗ ∼ 0, θ1 + θ1∗ → − ∞), (26d)

with qj1+’s and qj2+’s (j=1, 2, 3) as the asymptotic expressions for the vector two solitons after the collision, respectively.

Hereby, we set Δj1− (Δj1+) and Δj2− (Δj2+) (j = 1, 2, 3) as the amplitudes of the vector two solitons before (after) the collision. After some calculations, we obtain

(27)Δj1− = Δj1+ and Δj2− = Δj2+, j = 1, 2, 3, (27)

with the condition

(28)a1(1)a1(2) = a2(1)a2(2) = c, (28)

where c is a nonzero constant. When a1(j) and a2(j) (j = 1, 2) do not satisfy Condition (28), we have

(29a)Δj1− ≠ Δj1+ and Δj2− ≠ Δj2+, (29a)

(29b)Δ31− = Δ31+ and Δ32− = Δ32+. (29b)

From the asymptotic analysis earlier, we find that the appearance of the elastic or inelastic collision for the 2-bright-1-dark vector two solitons depends on the relation between a1(1)a1(2) and a2(1)a2(2). For the two bright solitons, elastic collision just occurs under Condition (28), while inelastic collision arises without Condition (28). However, collision for the dark solitons is always elastic.

Figures 7–10 show the collisions for the 2-bright-1-dark vector two solitons with the different velocities. In Figures 7 and 8, the choices of a1(1) and a2(2) satisfy Condition (28). In the homogeneous optical fibre where β(z) and γ(z) being the constants, such as β(z)=γ(z)=4, for the elastic collisions, amplitude and velocity for each soliton do not change before and after the collision except for a phase shift, as shown in Figure 7. For the inhomogeneous optical fibre, although the values of β(z) and γ(z) are variable in Figure 8, the value of β(z)γ(z) is a constant and the collisions are still elastic. Without Condition (28), inelastic collisions for the bright solitons occur in Figures 9a, b, d, e and 10a, b, d, e: After the collision, amplitudes of the solitons in the component q₁ increase, but they decrease in q₂. However, collisions for the dark solitons shown in Figures 9c, f and 10c, f are all elastic. Owing to the choices of β(z) and γ(z), the soliton velocities are invariable in Figure 9 but variable in Figure 10 during the propagation. One of the velocities of the vector two solitons shown in Figure 10 slowly decreases and eventually approaches a steady positive velocity, while the another one slowly increases and eventually approaches a steady negative velocity for t→+∞ or –∞.

$Figure 7: (a–c) Three-dimensional graphs of the elastic collision between the vector two solitons via Solutions (19) with a1(1) = 4,$a_1^{(1)}\, = \,4,$a2(1) = 2,$a_2^{(1)}\, = \,2,$a1(2) = 3,$a_1^{(2)}\, = \,3,$a2(2) = 1.5,$a_2^{(2)}\, = \,1.5,$χ=1.5+i, p=1, η1=0.6+i, η2=–1–i, β(z)=γ(z)=4. (d–f) Contour plots of (a–c), respectively.$

Figure 7:

(a–c) Three-dimensional graphs of the elastic collision between the vector two solitons via Solutions (19) with a1(1) = 4,a2(1) = 2,a1(2) = 3,a2(2) = 1.5,χ=1.5+i, p=1, η₁=0.6+i, η₂=–1–i, β(z)=γ(z)=4. (d–f) Contour plots of (a–c), respectively.

Figure 8:

The same as Figure 7 except that β(z)=γ(z)=4z².

$Figure 9: (a) and (b) Three-dimensional graphs of the inelastic collisions between the two bright solitons via Solutions (19). (c) Three-dimensional graph of the elastic collision between the two dark solitons via Solution (19). (d–f) Contour plots of (a–c), respectively. Hereby, the parameters are the same as those in Figure 7 except that a1(1) = 5,$a_1^{(1)}\, = \,5,$a2(1) = 1.2,$a_2^{(1)}\, = \,1.2,$a2(1) = 2,$a_2^{(1)}\, = \,2,$a2(2) = 3.$a_2^{(2)}\, = \,3.$$

Figure 9:

(a) and (b) Three-dimensional graphs of the inelastic collisions between the two bright solitons via Solutions (19). (c) Three-dimensional graph of the elastic collision between the two dark solitons via Solution (19). (d–f) Contour plots of (a–c), respectively. Hereby, the parameters are the same as those in Figure 7 except that a1(1) = 5,a2(1) = 1.2,a2(1) = 2,a2(2) = 3.

Figure 10:

The same as Figure 9 except β(z)=4z², γ(z)=4z².

(B) Asymptotic analysis on Solutions (22)

Before the collision (z→–∞),

(30a)q11− → β(z)γ(z)a1eΘ11 + σ1eΘ1 + Θ1∗, (Θ1 + Θ1∗ ∼ 0, Θ2 + Θ2∗ → − ∞), (30a)

(30b)qj1− → β(z)γ(z)χjeϕj[1 + κ1(j)eΘ1 + Θ1∗]1 + σ1eΘ1 + Θ1∗(j = 2, 3), (Θ1 + Θ1∗ ∼ 0, Θ2 + Θ2∗ → − ∞), (30b)

(30c)q12− → β(z)γ(z)A1eΘ2σ1 + veΘ2 + Θ2∗, (Θ2 + Θ2∗ ∼ 0, Θ1 + Θ1∗ → + ∞), (30c)

(30d)qj2− → β(z)γ(z)χjeϕj[κ1(j) + ζ(j)eΘ2 + Θ2∗]σ1 + veΘ2 + Θ2∗(j = 2, 3), (Θ2 + Θ2∗ ∼ 0, Θ1 + Θ1∗ → + ∞), (30d)

After the collision (z→+∞),

(31a)q11+ → β(z)γ(z)A2eΘ1σ4 + νeΘ1 + Θ1∗, (Θ1 + Θ1∗ ∼ 0, Θ2 + Θ2∗ → + ∞), (31a)

(31b)qj1+ → β(z)γ(z)χjeϕj[κ4(j) + ς(j)eΘ1 + Θ1∗]σ4 + veΘ1 + Θ1∗(j = 2, 3), (Θ1 + Θ1∗ ∼ 0, Θ2 + Θ2∗ → + ∞), (31b)

(31c)q12+ → β(z)γ(z)a2eθ21 + σ4eΘ2 + Θ2∗, (Θ2 + Θ2∗∼0, Θ1 + Θ1∗ → − ∞), (31c)

(31d)qj2+ → β(z)γ(z)χjeϕj[1 + κ4(j)eΘ2 + Θ2∗]1 + σ4eΘ2 + Θ2∗ (j = 2, 3), (Θ2 + Θ2∗ ∼ 0, Θ1 + Θ1∗ → − ∞), (31d)

We set Δ˜j1− (Δ˜j1+) and Δ˜j2− (Δ˜j2+) (j = 1, 2, 3) as the amplitudes of the vector two solitons before (after) the collision. After some calculations, we have

(32)Δ˜j1− = Δ˜j1+ and Δ˜j2− = Δ˜j2+. (32)

From Expression (32), we find that the collision for the 1-bright-2-dark vector two solitons is always elastic.

Figures 11 and 12 show the elastic collisions between the 1-bright-2-dark vector two solitons with the different velocities, for the homogeneous and inhomogeneous fibres, respectively. With β(z) and γ(z) being the constants, elastic collisions are shown in Figure 11. When the values of β(z) and γ(z) are variable and β(z)γ(z) is a constant, Figure 12 illustrates the different elastic collisions, where the soliton velocities are variable, and shows the same changing tendencies as those shown in Figure 10.

Figure 11:

(a–c) Three-dimensional graphs of the elastic collision between the vector two solitons via Solutions (22) with a₁=a₂=1, χ₂=1+i, χ₃=1–i, p₂=p₃=1, μ₁=0.5+i, μ₂=1–i, β(z)=γ(z)=2. (d–f) Contour plots of (a–c), respectively.

Figure 12:

The same as Figure 11 except that β(z)=γ(z)=z².

4.2.2 Bound States

In this part, taking the collisions between the 2-bright-1-dark vector two solitons as an example, we will investigate the effects of the GVD coefficient, β(z), and nonlinearity coefficient, γ(z), on the evolutions of the bound states via the graphic analysis. For the collisions between the 1-bright-2-dark vector two solitons, the analysis is similar.

Figures 13–17 show the 2-bright-1-dark vector two solitons in bound states with the invariable amplitudes, where the bound states shown in Figures 13–15 form in the homogeneous fibre, while those shown in Figures 16 and 17, in the inhomogeneous fibre. In Figures 13–15, two solitons in the components q₁, q₂, and q₃ all attract and repel each other periodically, where the values of β(z) and γ(z) are both constants. With the increase in β(z), the collision period of the solitons in Figure 14 is shorter than that in Figure 13. Comparing Figures 13 and 15, we find that the collision period is independent of γ(z). With the choices of nonconstant parameters β(z) and γ(z), the bound states shown in Figures 16 and 17 are different from those shown in Figures 13–15. In Figure 16, steady transmission for the vector two solitons can be achieved in a short distance at first, and then, soliton collision occurs periodically with a collision period becoming shorter and shorter. Figure 17 shows the vector two solitons attract and repel each other periodically except that the steady transmission occurs around z=0.

$Figure 13: (a–c) Three-dimensional graphs of the bound states via Solutions (19) with a1(1) = a2(1) = a1(2) = a2(2) = 1,$a_1^{(1)}\, = \,a_2^{(1)}\, = \,a_1^{(2)}\, = \,a_2^{(2)}\, = \,1,$χ=1.5+i, p=1, η1=1, η2=1.1, β(z)=γ(z)=10. (d–f) Contour plots of (a–c), respectively.$

Figure 13:

(a–c) Three-dimensional graphs of the bound states via Solutions (19) with a1(1) = a2(1) = a1(2) = a2(2) = 1,χ=1.5+i, p=1, η₁=1, η₂=1.1, β(z)=γ(z)=10. (d–f) Contour plots of (a–c), respectively.

Figure 14:

The same as Figure 13 except that β(z)=15.

Figure 15:

The same as Figure 13 except that γ(z)=25.

Figure 16:

The same as Figure 13 except that β(z)=γ(z)=10e^0.4z.

Figure 17:

The same as Figure 13 except that β(z)=γ(z)=4z.

5 Conclusions

In this article, we have investigated the propagation and collision of the vector solitons from the 3-coupled variable-coefficient NLS equations, i.e. (1), which describe the amplification or attenuation of the picosecond pulses in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations. On the basis of Lax Pair (3), Infinitely-Many Conservation Laws (8) have been derived. Under Constraint (2) among the variable coefficients for the GVD, β(z), nonlinearity, γ(z), and fibre gain/loss, δ(z), Bilinear Forms (16), two types of the analytic mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions, i.e. Solutions (18), (19), (21), and (22), have been obtained via the Hirota method and symbolic computation. On the basis of those solutions, influence of β(z) and γ(z) on the soliton propagation and collision has been analysed. With the asymptotic and graphic analysis, elastic and inelastic collisions between the vector two solitons have been studied. We have also analysed the effects of β(z) and γ(z) on the bound states via the graphic analysis. Attention should be paid to the following aspects:

Amplitudes of the bright and dark solitons are both related to β(z) and γ(z), while their velocities are only related to β(z).
With the choices of β(z) and γ(z), the vector soliton propagates in different forms, as seen in Figures 1, 2, 4, and 5. Besides, amplitude-changing optical vector soliton can be obtained when the value of β(z)γ(z) is variable, as seen in Figures 3 and 6.
For the 2-bright-1-dark vector two solitons, elastic collision for the two bright solitons only occurs with Condition (28), as seen in Figures 7a, b, d, e and 8a, b, d, e, whereas inelastic collision arises without Condition (28), as seen in Figures 9a, b, d, e and 10a, b, d, e. Collision for the dark solitons is always elastic, as seen in Figures 7c, f, 8c, f, 9c, f and 10c, f.
For the 1-bright-2-dark vector two solitons, collision in each component is elastic, as seen in Figures 11 and 12.
For the bound states, β(z) and γ(z) affect their types, as seen in Figures 13–17. Besides, with β(z) and γ(z) being the constants, collision period decreases as β(z) increases but is independent of γ(z), as seen in Figures 13–15.

Acknowledgments

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

References

[1] P. D. Green and A. Biswas, Commun. Nonlinear Sci. Numer. Simul. 15, 3865 (2010).10.1016/j.cnsns.2010.01.018Suche in Google Scholar

[2] A. Biswas and D. Milovic, Commun. Nonlinear Sci. Numer. Simul. 15, 1473 (2010).10.1016/j.cnsns.2009.06.017Suche in Google Scholar

[3] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).10.1063/1.1654836Suche in Google Scholar

[4] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).10.1103/PhysRevLett.45.1095Suche in Google Scholar

[5] M. S. Mani Rajana, A. Mahalingamb, and A. Uthayakumarc, Ann. Phys. 346, 1 (2014).10.1016/j.aop.2014.03.012Suche in Google Scholar

[6] C. Q. Su, Y. T. Gao, L. Xue, and X. Yu, Z. Naturforsch. A 70, 935 (2015).10.1515/zna-2015-0217Suche in Google Scholar

[7] T. Kanna and M. Lakshmanan, Phys. Rev. Lett. 86, 5043 (2001).10.1103/PhysRevLett.86.5043Suche in Google Scholar PubMed

[8] Q. M. Wang, Y. T. Gao, C. Q. Su, Y. J. Shen, Y. J. Feng, et al., Z. Naturforsch. A 70, 365 (2015).10.1515/zna-2015-0060Suche in Google Scholar

[9] H. Nakatsuka, D. Grischkowsky, and A. C. Balant, Phys. Rev. Lett. 47, 910 (1981).10.1103/PhysRevLett.47.910Suche in Google Scholar

[10] H. Bulut, Y. Pandir, and S. Tuluce Demiray, Waves Random Complex Media 24, 439 (2014).10.1080/17455030.2014.939246Suche in Google Scholar

[11] G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York 1995.Suche in Google Scholar

[12] C. R. Menyuk, Opt. Lett. 12, 614 (1987).10.1364/OL.12.000614Suche in Google Scholar

[13] C. R. Menyuk, J. Opt. Soc. Am. B 5, 392 (1988).10.1364/JOSAB.5.000392Suche in Google Scholar

[14] R. Radhakrishnan, P. T. Dinda, and G. Millot, Phys. Rev. E 69, 046607 (2004).10.1103/PhysRevE.69.046607Suche in Google Scholar PubMed

[15] R. Radhakrishnan, M. Lakshmanan, and J. Hietarinta, Phys. Rev. E 56, 2213 (1997).10.1103/PhysRevE.56.2213Suche in Google Scholar

[16] R. Radhakrishnan and K. Aravinthan, J. Phys. A 40, 13023 (2007).10.1088/1751-8113/40/43/011Suche in Google Scholar

[17] A. P. Sheppard and Y. S. Kivshar, Phys. Rev. E 55, 4473 (1997).10.1103/PhysRevB.55.4473Suche in Google Scholar

[18] X. Lü, J. Li, H. Q. Zhang, T. Xu, L. L. Li, et al. J. Math. Phys. 51, 043511 (2010).10.1063/1.3372723Suche in Google Scholar

[19] W. R. Shan, F. H. Qi, R. Guo, Y. S. Xue, P. Wang, et al. Phys. Scr. 85, 015002 (2012).10.1088/0031-8949/85/01/015002Suche in Google Scholar

[20] J. W. Yang, Y. T. Gao, Q. M. Wang, C. Q. Su, Y. J. Feng, et al., Physica B 481, 148 (2016).10.1016/j.physb.2015.10.025Suche in Google Scholar

[21] M. Hisakado and M. Wadati, J. Phys. Soc. Jpn. 64, 408 (1995).10.1143/JPSJ.64.408Suche in Google Scholar

[22] M. Hisakado, T. Iizuka, and M. Wadati, J. Phys. Soc. Jpn. 63, 2887 (1994).10.1143/JPSJ.63.2887Suche in Google Scholar

[23] Y. J. Feng, Y. T. Gao, Z. Y. Sun, D. W. Zuo, Y. J. Shen, et al., Phys. Scr. 90, 045201 (2015).10.1088/0031-8949/90/4/045201Suche in Google Scholar

[24] Q. M. Wang, Y. T. Gao, C. Q. Su, B. Q. Mao, Z. Gao, et al., Ann. Phys. 363, 440 (2015).10.1016/j.aop.2015.10.001Suche in Google Scholar

[25] C. Q. Dai and H. P. Zhu, Ann. Phys. 341, 142 (2014).10.1016/j.aop.2013.11.015Suche in Google Scholar

[26] V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).10.1103/PhysRevE.71.056619Suche in Google Scholar PubMed

[27] S. Kumar, K. Singh, and R. K. Gupta, Commun. Nonlinear Sci. Numer. Simul. 17, 1529 (2012).10.1016/j.cnsns.2011.09.003Suche in Google Scholar

[28] F. H. Qi, H. M. Ju, X. H. Meng, and J. Li, Nonlinear Dyn. 77, 1331 (2014).10.1007/s11071-014-1382-5Suche in Google Scholar

[29] X. W. Zhou and L. Wang, Comput. Math. Appl. 61, 2035 (2011).10.1016/j.camwa.2010.08.062Suche in Google Scholar

[30] H. J. Li, J. P. Tian, and L. J. Song, Optik 124, 7040 (2013).10.1016/j.ijleo.2013.05.137Suche in Google Scholar

[31] J. Tian and G. Zhou, Eur. Phys. J. D 41, 171 (2007).10.1140/epjd/e2006-00194-ySuche in Google Scholar

[32] V. R. Kumar, R. Radha, and K. Porsezian, Eur. Phys. J. D 57, 387 (2010).10.1140/epjd/e2010-00049-0Suche in Google Scholar

[33] R. Hirota, J. Math. Phys. 14, 805 (1973).10.1063/1.1666399Suche in Google Scholar

[34] R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge 2004.10.1017/CBO9780511543043Suche in Google Scholar

[35] B. Tian and Y. T. Gao, Phys. Lett. A 362, 283 (2007).10.1016/j.physleta.2006.10.094Suche in Google Scholar

[36] C. Q. Su, Y. T Gao, X. Yu, L. Xue, and Y. J. Shen, J. Math. Anal. Appl. 435, 735 (2016).10.1016/j.jmaa.2015.10.036Suche in Google Scholar

[37] P. Jin, C. A. Bouman, and K. D. Sauer, IEEE Trans. Comput. Imaging 1, 200 (2015).10.1109/TCI.2015.2461492Suche in Google Scholar

[38] K. Konno, H. Sanuki, and Y. H. Ichikawa, Prog. Theor. Phys. 52, 886 (1974).10.1143/PTP.52.886Suche in Google Scholar

[39] H. Sanuki and K. Konno, Phys. Lett. A 48, 221 (1974).10.1016/0375-9601(74)90553-2Suche in Google Scholar

Received: 2016-1-18

Accepted: 2016-3-18

Published Online: 2016-4-22

Published in Print: 2016-6-1

Artikel in diesem Heft

https://doi.org/10.1515/zna-2016-0019

Schlagwörter für diesen Artikel

3-Coupled Variable-Coefficient Nonlinear Schrödinger Equations; Infinitely-Many Conservation Laws; Mixed-Type Vector Soliton; Multicomponent Optical Fibre