Vector Dark Solitons for a Coupled Nonlinear Schrödinger System with Variable Coefficients in an Inhomogeneous Optical Fibre

Lei Liu; Bo Tian; Xiao-Yu Wu; Yu-Qiang Yuan

doi:10.1515/zna-2017-0148

Article

Vector Dark Solitons for a Coupled Nonlinear Schrödinger System with Variable Coefficients in an Inhomogeneous Optical Fibre

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Published/Copyright: July 19, 2017

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From the journal Zeitschrift für Naturforschung A Volume 72 Issue 8

Abstract

Studied in this paper are the vector dark solitons for a coupled nonlinear Schrödinger system with variable coefficients, which can be used to describe the pulse simultaneous propagation of the M-field components in an inhomogeneous optical fibre, where M is a positive integer. When M=2, under the integrable constraint, we construct the nondegenerate N-dark-dark soliton solutions in terms of the Gramian through the Kadomtsev–Petviashvili hierarchy reduction. With the help of analytic analysis, a vector one soliton with varying amplitude and velocity is studied. Interactions and bound states between the two solitons under different group velocity dispersion and amplification/absorption coefficients are presented. Moreover, we extend our analysis to any M to obtain the nondegenerate vector N-dark soliton solutions.

Keywords: Coupled Nonlinear Schrödinger System with Variable Coefficients; Inhomogeneous Optical Fibre; Kadomtsev–Petviashvili Hierarchy Reduction; Vector Dark Soliton

Acknowledgements

This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023 and No. 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.

Appendix A

In this Appendix, we will prove Theorem 1 in Section 2 via the KP hierarchy reduction. We present the solutions in terms of the Gramian for the bilinear equations in the KP hierarchy [31].

Lemma 1: Consider the following bilinear equations in the KP hierarchy [31]

(A.1a)(Dx2−Dx12−2αDx1)τn+1,k⋅τn,k=0,

(A.1b)(Dx1Dx−1(1)−2)τn,k⋅τn,k=−2τn+1,kτn−1,k.

(A.1c)(Dx2−Dx12−2βDx1)τn,k+1⋅τn,k=0,

(A.1d)(Dx1Dx−1(1)−2)τn,k⋅τn,k=−2τn,k+1τn,k−1.

where α and β are the complex constants, n and k are the integers, and τ_n,k is a function of four independent variables (x1, x2, x−1(1), x−1(2)). The determinant solution τ_n,k of the above equations is given by

(A.2)τn,k= |mlmn,k|1≤l,m≤N,

where the matrix element mlmn,k satisfies

(A.3a)∂x1mlmn,k=φln,kψmn,k,

(A.3b)∂x2mlmn,k=(∂x1φln,k)ψmn,k−φln,k(∂x1ψmn,k),

(A.3c)∂x−1(1)mlmn,k=−φln−1,kψmn+1,k, ∂x−1(2)mlmn,k=−φln,k−1ψmn,k+1,

(A.3d)mlmn+1,k=mlmn,k+φln,kψmn+1,k, mlmn,k+1=mlmn,k+φln,kψmn,k+1,

with φln,k and ψmn,k as the functions satisfying

(A.4a)∂x2φln,k=∂x12φln,k, ∂x2ψmn,k=−∂x12ψmn,k,

(A.4b)φln+1,k=(∂x1−α)φln,k, φln,k+1=(∂x1−β)φln,k,

(A.4c)ψmn−1,k=−(∂x1+α)ψmn,k, ψmn,k−1=−(∂x1+β)ψmn,k.

The proof of Lemma 1 has been given by the Gram technique [9], which is omitted here.

Then, based on Lemma 1, we define the matrix element mlmn,k by

(A.5a)mlmn,k=clm+1pl+ymφln,kψmn,k,

(A.5b)φln,k=(pl−α)n(pl−β)keξl,

(A.5c)ψmn,k=(−1ym+α)n(−1ym+β)keηm,

with

ξl=1pl−αx−1(1)+1pl−βx−1(2)+plx1+pl2x2+ξl(0),ηm=1ym+αx−1(1)+1ym+βx−1(2)+ymx1−ym2x2+ηm(0),

where c_lm, p_l, y_m, ξl(0), and ηm(0) are the complex constants. We can see that such mlmn,k,φln,k, and ψmn,k satisfy expressions (A.3) and (A.4). If we assume that x₁, x−1(1), and x−1(2) are real, x₂, α, and β are purely imaginary, and ym=pm∗,ηm(0)=ξm(0)∗,cml=clm∗, we have ηm=ξm∗,mmln,k=(mlm−n,−k)∗, and τn,k=τ−n,−k∗. Therefore, we can set

(A.6)clm=δlm, Re(pi)>0, f=τ0,0, g1=τ1,0, g2=τ0,1,

where δ_lm is the Kronecker delta notation. We have

(A.7a)(Dx2−Dx12−2αDx1)g1⋅f=0,

(A.7b)(Dx1Dx−1(1)−2)1f⋅f=−2|g1|2.

(A.7c)(Dx2−Dx12−2βDx1)g2⋅f=0,

(A.7d)(Dx1Dx−1(2)−2)f⋅f=−2|g2|2.

It is noted that bilinear forms (7) and (8) can be rewritten as

(A.8a)[−2iΛ(z)Dz−Dt2−2ib1Dt]g1⋅f=0,

(A.8b)[−2iΛ(z)Dz−Dt2−2ib2Dt]g2⋅f=0.

Therefore, we set

(A.9)x1=t, x2=12i∫Λ(z)dz, α=ib1, β=ib2,

and expressions (A.7a) and (A.7c) are reduced to bilinear forms (7) and (8), respectively.

In order to satisfy bilinear form (6), we consider the constraints

(A.10)−aσ1|μ1|2∂x−1()eξl+ξl∗−aσ2|μ2|2∂x−1()eξl+ξl∗=∂teξl+ξl∗.

i.e.

(A.11)σ1|μ1|2(pl−ib1)(pl∗+ib1)+σ2|μ2|2(pl−ib2)(pl∗+ib2)=−1a.

It is noted that

(A.12)f=|δlm+1pl+pm∗eξl+ξm∗|N×N=eξ1+…+ξN+ξ1∗+…+ξN∗|δlme−ξl−ξl∗+1pl+pm∗|N×N.

Then, we have

(A.13a)−aσ1|μ1|2fx−1(1)−aσ2|μ2|2fx−1(2)=ft,

(A.13b)−aσ1|μ1|2ftx−1(1)−aσ2|μ2|2ftx−1(2)=ftt.

Applying expressions (A.7b) and (A.7d), we obtain

(A.14)fttf−ft2+aσ1|μ1|2f2+aσ2|μ2|2f2= aσ1|μ1|2|g1|2+ aσ2|μ2|2|g2|2,

That is, bilinear form (6) is satisfied.

Therefore, under conditions (A.9) and (A.11), expression (A.7) is nothing but bilinear forms (6–8). Hereby the variables x−1(1) and x−1(2) just become two parameters to which we can substitute any values. Via expressions (A.5) and (A.6), setting x−1(1)=x−1(2)=0, we complete the proof of Theorem 1.

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Received: 2017-5-2

Accepted: 2017-6-10

Published Online: 2017-7-19

Published in Print: 2017-8-28

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Articles in the same Issue

https://doi.org/10.1515/zna-2017-0148

Keywords for this article

Coupled Nonlinear Schrödinger System with Variable Coefficients; Inhomogeneous Optical Fibre; Kadomtsev–Petviashvili Hierarchy Reduction; Vector Dark Soliton