N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation

In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi-discrete NLS equation; two-br...

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Дата:	2017
Автори:	Feng, B.-F., Ohta, Y.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2017
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/148759
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation / B.-F. Feng, Y. Ohta // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 51 назв. — англ.

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spelling	irk-123456789-1487592019-02-19T01:24:56Z N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation Feng, B.-F. Ohta, Y. In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi-discrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component semi-discrete NLS equation. The asymptotic behavior is analysed for two-soliton solutions. 2017 Article N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation / B.-F. Feng, Y. Ohta // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 51 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 39A10; 35Q55 DOI:10.3842/SIGMA.2017.071 http://dspace.nbuv.gov.ua/handle/123456789/148759 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi-discrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component semi-discrete NLS equation. The asymptotic behavior is analysed for two-soliton solutions.
format	Article
author	Feng, B.-F. Ohta, Y.
spellingShingle	Feng, B.-F. Ohta, Y. N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Feng, B.-F. Ohta, Y.
author_sort	Feng, B.-F.
title	N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation
title_short	N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation
title_full	N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation
title_fullStr	N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation
title_full_unstemmed	N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation
title_sort	n -bright-dark soliton solution to a semi-discrete vector nonlinear schrödinger equation
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148759
citation_txt	N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation / B.-F. Feng, Y. Ohta // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 51 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT fengbf nbrightdarksolitonsolutiontoasemidiscretevectornonlinearschrodingerequation AT ohtay nbrightdarksolitonsolutiontoasemidiscretevectornonlinearschrodingerequation
first_indexed	2025-07-12T20:10:44Z
last_indexed	2025-07-12T20:10:44Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 071, 16 pages N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation Bao-Feng FENG † and Yasuhiro OHTA ‡ † School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78539, USA E-mail: baofeng.feng@utrgv.edu ‡ Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: ohta@math.kobe-u.ac.jp Received April 25, 2017, in final form September 03, 2017; Published online September 06, 2017 https://doi.org/10.3842/SIGMA.2017.071 Abstract. In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota’s bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi- discrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component semi-discrete NLS equation. The asymptotic be- havior is analysed for two-soliton solutions. Key words: bright-dark soliton; semi-discrete vector NLS equation; Hirota’s bilinear method; Pfaffian 2010 Mathematics Subject Classification: 39A10; 35Q55 1 Introduction The nonlinear Schrödinger (NLS) equation iut = uxx + 2σ\|u\|2u is a generic model equation describing the evolution of small amplitude and slowly varying wave packets in weakly nonlinear media [2, 6, 8, 18, 23]. It arises in a variety of physical contexts such as nonlinear optics [19, 20], Bose–Einstein condensates [14], water waves [13] and plasma physics [48]. The integrability, as well as the bright-soliton solution in the focusing case (σ = 1), was found by Zakharov and Shabat [49, 50]. The dark soliton was found in the defocusing NLS equation (σ = −1) [20, 25, 42], and was observed experimentally in [24, 46]. The integrable discretization of nonlinear Schrödinger equation iqn,t = ( 1 + σ\|qn\|2 ) (qn+1 + qn−1) was originally derived by Ablowitz and Ladik [3, 4], so it is also called the Ablowitz–Ladik (AL) lattice equation. Similar to the continuous case, it is known that the AL lattice equation, by Hirota’s bilinear method, admits the bright soliton solution for the focusing case (σ = 1) [28, 41], also the dark soliton solution for the defocusing case (σ = −1) [27]. The inverse scattering transform (IST) has been developed by several authors in the literature [1, 36, 43, 44]. The geometric construction of the AL lattice equation was given by Doliwa and Santini [15]. This paper is a contribution to the Special Issue on Symmetries and Integrability of Difference Equations. The full collection is available at http://www.emis.de/journals/SIGMA/SIDE12.html mailto:baofeng.feng@utrgv.edu mailto:ohta@math.kobe-u.ac.jp https://doi.org/10.3842/SIGMA.2017.071 http://www.emis.de/journals/SIGMA/SIDE12.html 2 B.-F. Feng and Y. Ohta The coupled nonlinear Schrödinger equation iut = uxx + 2 ( σ1\|u\|2 + σ2\|v\|2 ) u, ivt = vxx + 2 ( σ1\|u\|2 + σ2\|v\|2 ) v, (1.1) where σi = ±1, i = 1, 2, was firstly recognized being integrable by Yajima and Oikawa [47]. For the focusing-focusing case (σ1 = σ2 = 1), the system (1.1) solved by Manakov via inverse scattering transform (IST), admits the bright-bright soliton solution [26], so it is also called the Manakov system in the literature. For the defocusing-defocusing case, the Manakov system admits bright-dark and dark-dark soliton solution [35, 37, 38]. However, the focusing-defocusing Manakov system admits all types of soliton solutions such as bright-bright solitons, dark-dark soliton, and bright-dark solitons [22, 33, 45]. The Manakov system can be easily extended to a multi-component case, the so-called vector NLS equation. For the continuous vector NLS equa- tion, the N -bright soliton solution was obtained in [16, 22, 51]; the general bright-dark and dark- dark soliton solutions were obtained in [16, 17, 25, 34, 45]. The inverse scattering transform with nonvanishing boundary condition was solved by Prinari, Ablowitz and Biondini [35]. We should remark here that the problem of constructing exact soliton solutions to the vector NLS equation and proving their nonsingularity was settled by Dubrovin et al. in their landmark paper [16]. The semi-discrete coupled nonlinear Schrödinger equation iq (1) n,t = ( 1 + σ1 ∣∣q(1) n ∣∣2 + σ2 ∣∣q(2) n ∣∣2)(q(1) n+1 + q (1) n−1 ) , iq (2) n,t = ( 1 + σ1 ∣∣q(1) n ∣∣2 + σ2 ∣∣q(2) n ∣∣2)(q(2) n+1 + q (2) n−1 ) , (1.2) where σi = ±1, i = 1, 2, is of importance both mathematically and physically. It was solved by the inverse scattering transform (IST) in [39, 40]. The general multi-soliton solution in terms of Pfaffians was found by one of the authors recently [31], which is of bright type for the focusing-focusing case (σ1 = σ2 = 1), and is of dark type for the defocusing-defocusing case (σ1 = σ2 = −1). However, as far as we know, no general mixed-type (bright-dark) soliton solution is reported in the literature, which motivated the present study. In the present paper, we consider a M -coupled semi-discrete NLS equation of all types iq (j) n,t = 1 + M∑ µ=1 σµ ∣∣q(µ) n ∣∣2(q(j) n+1 + q (j) n−1 ) , j = 1, 2, . . . ,M, (1.3) where σµ = ±1 for µ = 1, . . .M . For all-focusing case (σµ = 1, µ = 1, . . . ,M), its general N -bright soliton solution and the interactions among solitons were studied in [5, 29]. However, in contrast with a complete list of the general N -soliton solution to the vector NLS equation [17], the mixed-type soliton solution of all possible nonlinearities (all possible values of σµ) is missing. The aim of the present paper is to construct a general N -bright-dark soliton solution to the semi- discrete vector NLS equation. The rest of the paper is organized as follows. In Section 2, we provide a general bright-dark soliton solution in terms of Pfaffians to the semi-discrete vector NLS equation (1.3) and give a rigorous proof by the Pfaffian technique [21, 30, 32]. The one- and two-soliton solutions to two-coupled and three-coupled semi-discrete NLS equation are provided explicitly, respectively, in Section 3. We summarize the paper in Section 4 and present asymptotic analysis for two-soliton solution in Appendix A. 2 General bright-dark soliton solution to semi-discrete vector NLS equation Let us consider a general soliton solution consisting of m-bright solitons and (M − m)-dark solitons to the semi-discrete vector NLS equation (1.3). To this end, we introduce the following N -Bright-Dark Soliton Solution to a Vector SDNLS Equation 3 dependent variable transformations q(j) n = in g (j) n fn , q(m+l) n = ρl(ial) nh (l) n fn eωlt, (2.1) where j = 1, . . . ,m, ωl = s(al − āl), \|al\| = 1 with āl representing the complex conjugate of al, l = 1, . . . ,M −m. Here, fn is a real-valued function, whereas, gn and hn are complex-valued functions. The transformations convert equation (1.3) into a set of bilinear equations as follows Dtg (j) n · fn = s ( g (j) n+1fn−1 − g(j) n−1fn+1 ) , j = 1, . . . ,m, (Dt + ωl)h (l) n · fn = s ( alh (l) n+1fn−1 − ālh (l) n−1fn+1 ) , l = 1, . . . ,M −m, sfn+1fn−1 − f2 n = m∑ j=1 σj ∣∣g(j) n ∣∣2 + M−m∑ l=1 σl+m\|ρl\|2 ∣∣h(l) n ∣∣2. (2.2) Here s = 1 + M−m∑ l=1 σl+m\|ρl\|2. In what follows, we give a Pfaffian-type solution to the above bilinear equations. Theorem 2.1. A set of bilinear equations (2.2) admit the following solutions in the form of Pfaffians fn = Pf(a1, . . . , a2N , b1, . . . , b2N ), g(j) n = Pf(d0, βj , a1, . . . , a2N , b1, . . . , b2N ), h(l) n = Pf ( c (l) 1 , . . . , c (l) 2N , b1, . . . , b2N ) , with the elements of the Pfaffians defined as follows Pf(aj , ak)n = pj − pk pjpk − 1 ϕj(n)ϕk(n), Pf(d0, bj) = Pf(d0, βl) = 0, Pf ( c (l) j , c (l) k ) n = pj − pk pjpk − 1 pj − al 1− alpj pk − al 1− alpk ϕj(n)ϕk(n), Pf(aj , bk) = δjk, Pf ( c (l) j , bk ) = δjk, Pf(dl, ak)n = plkϕk(n), Pf(bj , βl) = { 0, 1 ≤ j ≤ N, α (l) j−N , N + 1 ≤ j ≤ 2N, Pf(aj , βl) = 0, Pf(bj , bk) = { bjk, 1 ≤ j ≤ N, N + 1 ≤ k ≤ 2N, 0, otherwise, with bjk = m∑ l=1 α (l) j σlα (l) k−N (pj − pk)(pjpk − 1) ( s pjpk − M−m∑ l=1 σl+m\|ρl\|2(al−āl)2 (1−alpj)(1−alpk)(1−ālpj)(1−ālpk) ) , and ϕj(n) = pnj e ηj , ηj = s ( pj − p−1 j ) t +ηj,0 which satisfying pj+N = p̄j, ηj+N,0 = η̄j,0. Proof. It can be shown that d dt fn = sPf(d−1, d1, . . . )n, where Pf(d−1, d1) ≡ 0, Pf(d±1, bj) ≡ 0, fn+1 = Pf(d0, d1, . . . )n, where Pf(d0, d1) ≡ 1, 4 B.-F. Feng and Y. Ohta fn−1 = Pf(d0, d−1, . . . )n, where Pf(d0, d−1) ≡ 1, and d dt g(j) n = sPf(d0, d−1, d1, βj , . . . )n, g (j) n+1 = Pf(d1, βj , . . . )n, g (j) n−1 = Pf(d−1, βj , . . . )n, where Pf(d0, d1, a1, . . . , a2N , b1, . . . , b2N ) is abbreviated by Pf(d0, d1, . . . ), so as other similar Pfaffians. Thus, an algebraic identity of Pfaffian Pf(d0, d−1, d1, βj , . . . )n Pf(. . . )n = Pf(d0, d−1, . . . )n Pf(d1, βj , . . . )n − Pf(d0, d1, . . . )n Pf(d−1, βj , . . . )n + Pf(d0, βj , . . . )n Pf(d−1, d1, . . . )n, together with above Pfaffian relations gives( d dt g(j) n ) × fn = sg (j) n+1 × fn−1 − sg(j) n−1 × fn+1 + g(j) n × ( d dt fn ) , which is exactly the first bilinear equation. Next we prove the second bilinear equation. It can also be shown that h(l) n = Pf ( d0, d̄ (l) 0 , . . . ) n , h (l) n+1 = āl Pf ( d1, d̄ (l) 0 , . . . ) n , h (l) n−1 = al Pf ( d−1, d̄ (l) 0 , . . . ) n ,( d dt + s(al − āl) ) h(l) n = sPf ( d0, d−1, d1, d̄ (l) 0 , . . . ) n , where Pf ( d̄ (l) 0 , aj ) = pnj ( pj − al 1− alpj ) eηj , Pf ( d̄ (l) 0 , bj ) = 0, Pf ( d0, d̄ (l) 0 ) = 1, Pf ( d−1,d̄ (l) 0 ) = āl, Pf ( d1, d̄ (l) 0 ) = al. Therefore, an algebraic identity of Pfaffian Pf ( d0, d−1, d1, d̄ (l) 0 , . . . ) n Pf(. . . )n = Pf(d0, d−1, . . . )n Pf ( d1, d̄ (l) 0 , . . . ) n − Pf(d0, d1, . . . )n Pf(d−1, d̄0, . . . )n + Pf ( d0, d̄ (l) 0 , . . . ) n Pf(d−1, d1, . . . )n, together with above Pfaffian relations gives( d dt + s(al − āl) ) h(l) n × fn = s ( alh (l) n+1fn−1 − ālh (l) n−1fn+1 ) + h(l) n ( d dt fn ) , which is nothing but the second bilinear equation. Now let us proceed to the proof of the third bilinear equation. To this end, we need to define Pf(aj , β̄l) = 0, Pf(bj , β̄l) = { 0, 1 ≤ j ≤ N, α (l) j−N , N + 1 ≤ j ≤ 2N, Pf ( d0, d′0 (l)) = 1, Pf ( c̄ (l) j , c̄ (l) k ) n = pj − pk pjpk − 1 1− alpj pj − al 1− alpk pk − al ϕj(n)ϕk(n), Pf ( d′0 (l) , aj ) = pnj ( 1− alpj pj − al ) eηj , Pf ( c̄ (l) j , bk ) = δjk, Pf ( d′0 (l) , bj ) = 0. N -Bright-Dark Soliton Solution to a Vector SDNLS Equation 5 Then from the fact Pf(aj , ak) = Pf(aj′ , ak′), Pf(bj , bk) = Pf(bj′ , bk′), where j′ = j +N mod(2N), k′ = k +N mod(2N), we obtain f̄n = fn, ḡ(j) n = Pf(d0, β̄j , a1, . . . , a2N , b1, . . . , b2N )n, h̄(l) n = Pf ( c̄ (l) 1 , . . . , c̄ (l) 2N , b1, . . . , b2N ) = Pf ( d0, d′0 (l) , . . . ) n . Since fn+1 = Pf(d0, d1, . . . )n, fn−1 = Pf(d0, d−1, . . . )n, we then have fn+1 = fn + 2N∑ j=1 (−1)j−1 Pf(d1, aj) Pf(d0, . . . , âj , . . . )n, fn−1 = fn + 2N∑ j=1 (−1)j−1 Pf(d−1, aj) Pf(d0, . . . , âj , . . . )n, Then we can show fn+1fn−1 − fnfn = − ∑ j<k (−1)j+k ( pj + 1 pj − pk − 1 pk ) Pf(bj,bk) × Pf(d0, . . . , b̂j , . . . ) Pf(d0, . . . , b̂k, . . . ). On the other hand, since h(l) n = Pf ( d0, d0 (l) , . . . ) n , h̄(l) n = Pf ( d0, d′0 (l) , . . . ) n , we have h(l) n = fn + 2N∑ j=1 (−1)j−1 Pf(d0, aj)n ( pj − al 1− alpj ) Pf(d0, . . . , âj , . . . )n, h̄(l) n = fn + 2N∑ j=1 (−1)j−1 Pf(d0, aj)n ( 1− alpj pj − al ) Pf(d0, . . . , âj , . . . )n. Similarly, we can show∣∣h(l) n ∣∣2 − fnfn = − ∑ j<k (−1)j+k ( pj − al 1− alpj + 1− alpj pj − al − pk − al 1− alpk − 1− alpk pk − al ) × Pf(bj,bk) Pf(d0, . . . , b̂j , . . . ) Pf(d0, . . . , b̂k, . . . ). Finally, we have s(fn+1fn−1 − fnfn)− M−m∑ l=1 σl+m\|ρl\|2 ( \|h(l) n \|2 − fnfn ) = − ∑ j<k (−1)j+k { s ( pj + 1 pj − pk − 1 pk ) 6 B.-F. Feng and Y. Ohta − M−m∑ l=1 σl+m\|ρl\|2 ( pj − al 1− alpj + 1− alpj pj − al − pk − al 1− alpk − 1− alpk pk − al )} × Pf(bj,bk) Pf(d0, . . . , b̂j , . . . ) Pf(d0, . . . , b̂k, . . . ) = N∑ j=1 2N∑ k=N+1 (−1)j+k m∑ l=1 α (l) j σlα (l) k−N Pf(d0, . . . , b̂j , . . . ) Pf(d0, . . . , b̂k, . . . ) = m∑ l=1 σl N∑ j=1 2N∑ k=N+1 (−1)j+k Pf(bj , βl) Pf(bk, β̄l) Pf(d0, . . . , b̂j , . . . ) Pf(d0, . . . , b̂k, . . . ) = m∑ l=1 σl\|g(l) n \|2. The third bilinear equation is proved. � The above Pfaffian solutions, with dependent variable transformations (2.1), give general N -bright-dark soliton solutions to the semi-discrete vector NLS equation (1.3). 3 One- and two-soliton solutions for the two- and three-coupled discrete NLS equation 3.1 Two-component semi-discrete NLS equation In this subsection, we provide and illustrate one- and two-soliton for two-component semi- discrete NLS equation (1.2) explicitly. One-soliton solution. the tau-functions for one-soliton solution (N = 1) are fn = −1− c11̄(p1p̄1)neη1+η̄1 , g(1) n = −α(1) 1 p1 neη1 , h(1) n = −1− d(1) 11̄ (p1p̄1)neη1+η̄1 , where c11̄ = ᾱ (1) 1 σ1α (1) 1 (p1p̄1 − 1)2 ( s \|p1\|2 + σ2\|ρ1\|2(a1−ā1)2 \|1−a1p1\|2\|1−ā1p1\|2 ) , d (1) 11̄ = − (p1 − a1)(p̄1 − a1) (1− a1p1)(1− a1p̄1) c11̄. The above tau functions lead to the one-soliton solution as follows q(1) n = α (1) 1 2 √ c11̄ eiξ1I sech(ξ1R + θ0), q(2) n = 1 2 ρ1e iζ1 ( 1 + e2iφ1 + (e2iφ1 − 1) tanh(ξ1R + θ0) ) , where ξ1 = ξ1R + iξ1I = n ln(ip1) + s ( p1 − p−1 1 ) t, ζ1 = nϕ1 + nπ/2 + ω1t, e iϕ1 = a1, e2θ0 = c11̄, e2iφ1 = −(p1 − a1)(p̄1 − a1)/((1 − a1p1)(1 − a1p̄1)). Therefore, the amplitude of bright soliton for q(1) are 1 2 ∣∣α(1) 1 ∣∣/√a11̄. The dark soliton q(2) approaches \|ρ1\| as x → ±∞. In addition, the intensity of the dark soliton is \|ρ1\| cosφ1. An example of one-bright-dark soliton is illustrated in Fig. 1a p1 = 1.0 + 0.8i, ρ1 = 5.0, α (1) 1 = 1.0 + i, a1 = 0.8 + 0.6i. for focusing-focusing case (σ1 = 1.0, σ2 = 1.0). Fig. 1b shows the bright-dark soliton for the focusing-defocusing case (σ1 = 1.0, σ2 = −1.0). It is interesting to note the dark soliton corresponding to defocusing component becomes an anti-dark one. N -Bright-Dark Soliton Solution to a Vector SDNLS Equation 7 −50 0 50 0 2 4 6 8 10 n \|q(1) n \| \|q(2) n \| −50 0 50 0 5 10 15 n \|q(1) n \| \|q(2) n \| Figure 1. One-bright-dark soliton soliton solution to a two-coupled semi-discrete NLS equation: (a) focusing-focusing case (σ1 = 1.0, σ2 = 1.0); (b) focusing-defocusing case (σ1 = 1.0, σ2 = −1.0). −500 0 500 0 1 2 3 4 n \|q(1) n \| \|q(2) n \| −500 0 500 0 1 2 3 4 n \|q(1) n \| \|q(2) n \| Figure 2. A two-soliton solution of mixed type for a two-coupled semi-discrete NLS equation: (a) before the collision t = −80; (b) after the collision t = 80. Two-soliton solution. The tau functions for two-soliton are of the following form fn = 1 + c11̄E1Ē1 + c21̄E2Ē1 + c12̄E1Ē2 + c22̄E2Ē2 + c121̄2̄E1E2Ē1Ē2, gn = α (1) 1 E1 + α (1) 2 E2 + c (j) 121̄ E1E2Ē1 + c (j) 122̄ E1E2Ē2, hn = 1 + d (1) 11̄ E1Ē1 + d (1) 21̄ E2Ē1 + d (1) 12̄ E1Ē2 + d (1) 22̄ E2Ē2 + d (1) 121̄2̄ E1E2Ē1Ē2, where Ej = pj neηj , cij̄ = ᾱ (1) i σ1α (1) j (pip̄j − 1)2 ( s pip̄j + σ2\|ρ1\|2(a1−ā1)2 \|1−a1pi\|2\|1−ā1pj \|2 ) , d (1) ij̄ = − (pi − a1)(p̄j − a1) (1− a1pi)(1− a1p̄j) cij̄ , c121̄2̄ = \|p2 − p1\|2 ( c11̄c22̄ (p1 + p̄2) (p2 + p̄1) − c12̄c21̄ (p1 + p̄1)(p2 + p̄2) ) , c12j̄ = (p2 − p1) ( α (1) 2 c1j̄ p2 + p̄j − α (1) 1 c2j̄ p1 + p̄j ) , d121̄2̄ = (p1 − a1)(p̄1 − a1)(p2 − a2)(p̄1 − a2) (1− a1p1)(1− a1p̄1)(1− a1p2)(1− a1p̄2) c121̄2̄. A two bright-dark soliton solution is shown in Fig. 2 before and after the collision for parameters σ1 = 1.0, σ2 = −1, p1 = 1.0 + 0.5i, ρ = 2.0, α (1) 1 = 1.0 + 0.8i, α (1) 2 = 1.0 + 0.5i, a1 = 0.6 + 0.8i. It can be seen that the collision is elastic which is the same as for the continuous two-component NLS equation. 8 B.-F. Feng and Y. Ohta 3.2 Bright-dark soliton solution for three-component semi-discrete NLS equation In this subsection, we will give the mixed-type soliton solution to the following three-component semi-discrete NLS equation iq (1) n,t = ( 1 + 3∑ k=1 σk ∣∣q(k) n ∣∣2)(q(1) n+1 + q (1) n−1 ) , iq (2) n,t = ( 1 + 3∑ k=1 σk ∣∣q(k) n ∣∣2)(q(2) n+1 + q (2) n−1 ) , iq (3) n,t = ( 1 + 3∑ k=1 σk ∣∣q(k) n ∣∣2)(q(3) n+1 + q (3) n−1 ) . Two-bright-one-dark soliton solution. The tau-functions for one-soliton solution (N = 1) are fn = −1− c11̄(p1p̄1)neη1+η̄1 , g(j) n = −α(j) 1 p1 neη1 , h(1) n = −1− d(1) 11̄ (p1p̄1)neη1+η̄1 , where c11̄ = 2∑ k=1 ᾱ (k) 1 σkα (k) 1 (p1p̄1 − 1)2 ( s \|p1\|2 + σ3\|ρ1\|2(a1−ā1)2 \|1−a1p1\|2\|1−ā1p1\|2 ) , d (1) 11̄ = − (p1 − a1)(p̄1 − a1) (1− a1p1)(1− a1p̄1) c11̄, where ω = s(a1− ā1), \|a1\| = 1, s = 1+σ3\|ρ1\|2 with ā1 representing the complex conjugate of a1. The above tau functions lead to the one-soliton as follows q(j) n = α (j) 1 2 √ c11̄ eiξ1I sech(ξ1R + θ0), j = 1, 2 q(3) n = 1 2 ρ1e iζ1 ( 1 + e2iφ1 + (e2iφ1 − 1) tanh(ξ1R + θ0) ) , where ξ1 = ξ1R + iξ1I = n ln(ip1) + s ( p1 − p−1 1 ) t, ζ1 = nϕ1 + nπ/2 + ω1t, e iϕ1 = a1, e2θ0 = c11̄, e2iφ1 = −(p1 − a1)(p̄1 − a1)/((1 − a1p1)(1 − a1p̄1)). Therefore, the amplitude of bright soliton for q(j) are 1 2 ∣∣α(j) 1 ∣∣/√a11̄. The dark soliton q(3) approaches \|ρ1\| as x → ±∞. In addition, the intensity of the dark soliton is \|ρ1\| cosφ1. The tau functions for two-soliton are of the following form fn = 1 + c11̄E1Ē1 + c21̄E2Ē1 + c12̄E1Ē2 + c22̄E2Ē2 + c121̄2̄E1E2Ē1Ē2, g(j) n = α (j) 1 E1 + α (j) 2 E2 + c (j) 121̄ E1E2Ē1 + c (j) 122̄ E1E2Ē2, j = 1, 2, h(1) n = 1 + d (1) 11̄ E1Ē1 + d (1) 21̄ E2Ē1 + d (1) 12̄ E1Ē2 + d (1) 22̄ E2Ē2 + d (1) 121̄2̄ E1E2Ē1Ē2, where cij̄ = 2∑ k=1 ᾱ (k) i σkα (k) j (pip̄j − 1)2 ( s pip̄j + σ3\|ρ1\|2(a1−ā1)2 \|1−a1pi\|2\|1−ā1pj \|2 ) , d (1) ij̄ = − (pi − a1)(p̄j − a1) (1− a1pi)(1− a1p̄j) cij̄ , c121̄2̄ = \|p2 − p1\|2 ( c11̄c22̄ (p1 + p̄2)(p2 + p̄1) − c12̄c21̄ (p1 + p̄1)(p2 + p̄2) ) , N -Bright-Dark Soliton Solution to a Vector SDNLS Equation 9 c12j̄ = (p2 − p1) ( α (1) 2 c1j̄ p2 + p̄j − α (1) 1 c2j̄ p1 + p̄j ) , d (1) 121̄2̄ = (p1 − a1)(p̄1 − a1)(p2 − a2)(p̄1 − a2) (1− a1p1)(1− a1p̄1)(1− a1p2)(1− a1p̄2) c121̄2̄. It is found that two-bright-one-dark soliton solution given above is nonsingular for any other combinations of nonlinearities if the following quantity( 2∑ k=1 ᾱ (k) i σkα (k) j )( s \|pi\|2 + σ3\|ρ1\|2(a1 − ā1)2 \|1− a1pi\|2\|1− ā1pi\|2 ) is positive. One-bright–two-dark soliton solution. The tau-functions for one-soliton solution (N = 1) are fn = −1− c11̄(p1p̄1)neη1+η̄1 , g(1) n = −α(1) 1 p1 neη1 , h(l) n = −1− d(l) 11̄ (p1p̄1)neη1+η̄1 , where c11̄ = ᾱ (1) 1 σ1α (1) 1 (p1p̄1 − 1)2  s \|p1\|2 + 2∑ l=1 σl+1\|ρl\|2(al−āl)2 \|1−a1p1\|2\|1−ā1p1\|2  . Here ωl = s(al − āl), \|al\| = 1, s = 1 + σ2\|ρ1\|2 + σ3\|ρ2\|2, āl represents the complex conjugate of al. The above tau functions lead to the one-soliton as follows q(1) n = α (1) 1 2 √ c11̄e iξ1I sech(ξ1R + θ0), q(l+1) n = 1 2 ρle iζl ( 1 + e2iφl + (e2iφl − 1) tanh(ξ1R + θ0) ) , l = 1, 2, where ξ1 = ξ1R + iξ1I = n ln(ip1) + s ( p1 − p−1 1 ) t, ζl = nϕl + nπ/2 + ωlt, e iϕl = al, e 2θ0 = c11̄, e2iφl = −(p1 − al)(p̄1 − al)/((1 − alp1)(1 − alp̄1)). Therefore, the amplitude of bright soliton for q(1) are 1 2 ∣∣α(1) 1 ∣∣/√a11̄. The dark soliton q(l+1) approaches \|ρl\| as x → ±∞. In addition, the intensity of the dark soliton is \|ρl\| cosφl. The tau functions for two-soliton are of the following form fn = 1 + c11̄E1Ē1 + c21̄E2Ē1 + c12̄E1Ē2 + c22̄E2Ē2 + c121̄2̄E1E2Ē1Ē2, g(1) n = α (1) 1 E1 + α (1) 2 E2 + c (1) 121̄ E1E2Ē1 + c (1) 122̄ E1E2Ē2, h(l) n = 1 + d (l) 11̄ E1Ē1 + d (l) 21̄ E2Ē1 + d (l) 12̄ E1Ē2 + d (l) 22̄ E2Ē2 + d (l) 121̄2̄ E1E2Ē1Ē2, l = 1, 2, where cij̄ = ᾱ (1) i σ1α (1) j (pip̄j − 1)2  s pip̄j + 2∑ l=1 σl+1\|ρl\|2(al−āl)2 \|1−alpi\|2\|1−ālpj \|2  , d (l) ij̄ = − (pi − al)(p̄j − al) (1− alpi)(1− alp̄j) cij̄ , c121̄2̄ = \|p2 − p1\|2 ( c11̄c22̄ (p1 + p̄2)(p2 + p̄1) − c12̄c21̄ (p1 + p̄1)(p2 + p̄2) ) , 10 B.-F. Feng and Y. Ohta c12j̄ = (p2 − p1) ( α (1) 2 c1j̄ p2 + p̄j − α (1) 1 c2j̄ p1 + p̄j ) , d (l) 121̄2̄ = (p1 − al)(p̄1 − al)(p2 − al)(p̄1 − al) (1− alp1)(1− alp̄1)(1− alp2)(1− alp̄2) c121̄2̄. For all possible combinations of mixed type in three-coupled NLS equation, one-bright-two- dark soliton solution exists if the following quantity ( ᾱ (1) i σ1α (1) i )  s \|pi\|2 + 2∑ l=1 σl+1\|ρl\|2(al − āl)2 \|1− alpi\|2\|1− ālpi\|2  is positive. The asymptotic analysis for two-soliton solution is performed in Appendix A. It should be pointed out that two-soliton for one-bright-two-dark soliton case always undertakes elastic col- lision without shape changing. 4 Discussion and conclusion We conclude the present paper by two comments. First, we comment on a connection of the vector semi-discrete NLS equation to the vector modified Volterra lattice equation studied in [7]. To this end, we consider the two-component semi-discrete NLS equation of focusing type i d dt un = ( 1 + \|un\|2 + \|vn\|2 ) (un+1 + un−1), i d dt vn = ( 1 + \|un\|2 + \|vn\|2 ) (vn+1 + vn−1). (4.1) Let un = xn + iyn and vn = zn + iwn, then equation (4.1) becomes d dt xn = ( 1 + x2 n + y2 n + z2 n + w2 n ) (yn+1 + yn−1), − d dt yn = ( 1 + x2 n + y2 n + z2 n + w2 n ) (xn+1 + xn−1), d dt zn = ( 1 + x2 n + y2 n + z2 n + w2 n ) (wn+1 + wn−1), − d dt wn = ( 1 + x2 n + y2 n + z2 n + w2 n ) (zn+1 + zn−1). By defining U (1) n = { xn, for n even, yn, for n odd, U (2) n = { −yn, for n even, xn, for n odd, U (3) n = { zn, for n even, wn, for n odd, U (4) n = { −wn, for n even, zn, for n odd, U (5) n = { (−1)n/2, for n even, (−1)(n−1)/2, for n odd, we obtain d dt U (1) n = (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (1) n+1 + U (1) n−1 ) , N -Bright-Dark Soliton Solution to a Vector SDNLS Equation 11 d dt U (2) n = (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (2) n+1 + U (2) n−1 ) , d dt U (3) n = (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (3) n+1 + U (3) n−1 ) , d dt U (4) n = (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (4) n+1 + U (4) n−1 ) , d dt U (5) n = (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (5) n+1 + U (5) n−1 ) for n being even, and d dt U (1) n = − (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (1) n+1 + U (1) n−1 ) , d dt U (2) n = − (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (2) n+1 + U (2) n−1 ) , d dt U (3) n = − (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (3) n+1 + U (3) n−1 ) , d dt U (4) n = − (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (4) n+1 + U (4) n−1 ) , d dt U (5) n = − (( U (1) n )2 + ( U (2) n )2 + ( U (3) n )2 + ( U (4) n )2 + ( U (5) n )2)( U (5) n+1 + U (5) n−1 ) for n being odd, in other words, d dt U (j) n = (−1)n ( 5∑ k=1 ( U (k) n )2)( U (j) n+1 + U (j) n−1 ) for 1 ≤ j ≤ 5. By rewriting U (j) n = inV (j) n and t = x/i, we have d dx V (j) n = ( 5∑ k=1 ( V (k) n )2)( V (j) n+1 − V (j) n−1 ) . Consequently, un = { U (1) n − iU (2) n , n even, U (2) n + iU (1) n , n odd = { in−1 ( V (2) n + iV (1) n ) , n even, in ( V (2) n + iV (1) n ) , n odd, vn = { U (3) n − iU (4) n , n even, U (4) n + iU (3) n , n odd = { in−1 ( V (4) n + iV (3) n ) , n even, in ( V (4) n + iV (3) n ) , n odd. un and u∗n correspond to U (1) n and U (2) n , and vn and v∗n correspond to U (3) n and U (4) n , with even-odd parity depending gauge factor i(n or n− 1). The correspondence between coupled defocusing-defocusing and focusing-defocusing coupled Ablowitz–Ladik equation and the vector modified Volterra lattice equatino can be constructed by similar variable transformations. In all cases, we need 5-components in the vector modified Volterra lattice, one of which is a trivial wave V (5) n = i(0 or 1), in order to recover 2-component coupled Ablowitz–Ladik equation. In principle any solutions of coupled Ablowitz–Ladik equation can be derived from those of vector modified Volterra lattice and vice versa. However it is not so easy to make exact matching. Second, we give a comparison between the NLS-type equations and the sine/sinh-Gordon- type equations since their belong to the simplest positive and negative flows of the AKNS hierarchy, respectively. In a series papers by Barashenlov, Getmanov et al. [9, 10, 11, 12], 12 B.-F. Feng and Y. Ohta a generic integrable relativistic system associated with the sl(2,C) was systematically investi- gated, from which the massive Thirring model, the complex sine-Gordon equation in Euclidean and Minkowski spaces etc. are produced by different reductions. Especially an O(1, 1) sine- Gordon equation with the Lagrangian L = u1ξu1η − u2ξu2η 1− ( u2 1 − u2 2 ) + ( u2 1 − u2 2 − 1 ) , exhibits nontrivial interaction of solitons such as decay and fusion [10]. Here u1 and u2 are real variables. In parallel to the O(1, 1) sine-Gordon equation, we can have a NLS equation of O(1, 1) type, whose Lagrangian is L = u1u2t − u1tu2 + ( u2 1x − u2 2x ) − σ ( u2 1 − u2 2 )2 . On the contrary, in parallel to the U(1, 1) coupled NLS system, e.g., system (1.1) with σ = 1 and σ = −1, whose Lagrangian can be written by L = i 2 ( utu ∗ − u∗ut − vtv∗ + v∗vt ) + ( \|u\|2x − \|v\|2x ) − σ ( \|u\|2 − \|v\|2 )2 , where u and v are complex variable and ∗ represents complex conjugate, we could propose a U(1, 1) coupled complex sine-Gordon system with Lagrangian L = uηu ∗ ξ − vηv∗ξ 1− ( \|u\|2 − \|v\|2 ) − (\|u\|2 − \|v\|2). Then several natural questions arise: does the NLS equation of O(1, 1) type exhibit nontrivial interaction of solitons such as decay and fusion? Are there any nontrivial interaction of solitons for the U(1, 1) coupled NLS system and the U(1, 1) complex sine-Gordon system? Unfortunately, the answers to these questions are not clear at this moment. We expect above questions could be answered by the authors or others in the near future. A Appendix By taking N = 1 we get the tau functions for one-soliton solution, fn = Pf(a1, a2, b1, b2) = −1− c11̄E1Ē1, g(j) n = Pf(d0, βj , a1, a2, b1, b2) = −α(1) 1 E1, j = 1, . . . ,m, h(l) n = Pf ( c (l) 1 , c (l) 2 , b1, b2 ) = −1− d(l) 11̄ E1Ē1, l = 1, . . . ,M −m, where c11̄ = m∑ k=1 ᾱ (k) 1 σkα (k) 1 (p1p̄1 − 1)2 ( s \|p1\|2 + M−m∑ l=1 \|ρl\|2(al−āl)2 \|1−alp1\|2\|1−ālp1\|2 ) , d (l) 11̄ = − (p1 − al)(p̄1 − al) (1− alp1)(1− alp̄1) c11̄. Based on the N -soliton solution of the vector discrete NLS equation, the tau-functions for two-soliton solution can be expanded for N = 2 f = Pf(a1, a2, a3, a4, b1, b2, b3, b4) = 1 + c11̄E1Ē1 + c21̄E2Ē1 + c12̄E1Ē2 + c22̄E2Ē2 + c121̄2̄E1E2Ē1Ē2, N -Bright-Dark Soliton Solution to a Vector SDNLS Equation 13 g(j) n = Pf(d0, βj , a1, a2, a3, a4, b1, b2, b3, b4) = α (j) 1 E1 + α (j) 2 E2 + c (j) 121̄ E1E2Ē1 + c (j) 122̄ E1E2Ē2, j = 1, . . . ,m, h(l) n = Pf ( c (l) 1 , c (l) 2 , c (l) 3 , c (l) 4 , b1, b2, b3, b4 ) = 1 + d (l) 11̄ E1Ē1 + d (l) 21̄ E2Ē1 + d (l) 12̄ E1Ē2 + d (l) 22̄ E2Ē2 + d (l) 121̄2̄ E1E2Ē1Ē2, l = 1, . . . ,M −m, where cij̄ = m∑ k=1 ᾱ (k) j σkα (k) i (pip̄j − 1)2 ( s pip̄j + M−m∑ l=1 ρl\|2(al−āl)2 \|1−alpi\|2\|1−ālpj \|2 ) , d (l) ij̄ = − (pi − al)(p̄j − al) (1− alpi)(1− alp̄j) cij̄ , c121̄2̄ = \|P12\|2 ( P11̄P22̄c12̄c21̄ − P12̄P21̄c11̄c22̄ ) , c (k) 12j̄ = P12 ( α (k) 1 P11̄c2j̄ − α (k) 2 P21̄c1j̄ ) , d (l) 121̄2̄ = (p1 − al)(p̄1 − al)(p2 − al)(p̄1 − al) (1− alp1)(1− alp̄1)(1− alp2)(1− alp̄2) c121̄2̄, Pij = pi − pj pipj − 1 , Pij̄ = pi − p̄j pip̄j − 1 . Next, we investigate the asymptotic behavior of two-soliton solution. To this end, we assume (Re ln(ip2)) > Re(ln(ip1)) > 0, sRe(p2−p−1 2 )/Re(ln(ip2)) > sRe(p1−p−1 1 )/Re(ln(ip1)) without loss of generality. For the above choice of parameters, we have (i) η1R ≈ 0, η2R → ∓∞ as t → ∓∞ for soliton 1 and (ii) η2R ≈ 0, η1R → ±∞ as t → ∓∞ for soliton 2. This leads to the following asymptotic forms for two-soliton solution. (i) Before collision (t→ −∞). Soliton 1 (η1R ≈ 0, η2R → −∞): q(j) n → α (j) 1 inE1 1 + c11̄E1Ē1 → A1− j eiξ1I sech ( ξ1R + ξ1− 0 ) , q(l+m) n → 1 + d (l) 11̄ E1Ē1 1 + c11̄E1Ē1 ρl(ial) neωlt → 1 2 ρle iζl ( 1 + e2iφ (1) l + ( e2iφ (1) l − 1 ) tanh ( ξ1R + ξ1− 0 )) , where A1− j = α (j) 1 2 √ c11̄ , e2ξ1−0 = c11̄, e2iφ (1) l = − (p1 − al)(p̄1 − al) (1− alp1)(1− alp̄1) , ζl = nϕl + nπ/2 + ωlt, ξ1 = ξ1R + iξ1I = n ln(ip1) + s ( p1 − p−1 1 ) t, eiϕl = al. Soliton 2 (η2R ≈ 0, η1R →∞): q(j) n → inc (j) 121̄ E2 c11̄ + c121̄2̄E2Ē2 → A2− j eiξ2I sech ( ξ2R + ξ2− 0 ) , q(l+m) n → d (l) 11̄ + d (l) 121̄2̄ E2Ē2 c11̄ + c121̄2̄E2Ē2 ρl(ial) neωlt → 1 2 ρle i ( ζl+2φ (1) l )( 1 + e2iφ (2) l + ( e2iφ (2) l − 1 ) tanh ( ξ2R + ξ2− 0 )) , 14 B.-F. Feng and Y. Ohta where A2− j = c (j) 121̄ 2 √ c11̄ √ c121̄2̄ , e2ξ2−0 = c121̄2̄ c11̄ , e2iφ (2) l = − (p2 − al)(p̄2 − al) (1− alp2)(1− alp̄2) , ζ2− 0 = nϕl + nπ/2 + ωlt, ξ2 = ξ2R + iξ2I = n ln(ip2) + s ( p2 − p−1 2 ) t. (ii) After the collision (t→∞). Soliton 1 (η1R ≈ 0, η2R → +∞): q(j) n → c (j) 122̄ inE1 c22̄ + c121̄2̄E1Ē1 → A1+ j eiξ1I sech ( ξ1R + ξ1+ 0 ) , q(l+m) n → d (l) 22̄ + d (l) 121̄2̄ E1Ē1 c22̄ + c121̄2̄E1Ē1 ρl(ial) neωlt → 1 2 ρle i ( ζl+2iφ (2) l )( 1 + e2iφ (1) l + ( e2iφ (1) l − 1 ) tanh ( ξ1R + ξ1+ 0 )) , where A1+ j = c (j) 122̄ 2 √ c11̄ √ c121̄2̄ , e2ξ1+0 = c121̄2̄ c22̄ , e2iφ (2) l = − (p2 − al)(p̄2 − al) (1− alp2)(1− alp̄2) , ζl = nϕl + nπ/2 + 2φ (2) l + ωlt, Soliton 2 (η2R ≈ 0, η1R → −∞): q(j) n → α (j) 2 inE2 1 + c22̄E2Ē2 → A2+ j eiξ2I sech ( ξ2R + ξ2+ 0 ) , q(l+m) n → 1 + d (l) 22̄ E2Ē2 1 + c22̄E2Ē2 ρl(ial) neωlt → 1 2 ρle iζl ( 1 + e2iφ (2) l + ( e2iφ (2) l − 1 ) tanh ( ξ2R + ξ2+ 0 )) , where A2+ j = α (j) 2 2 √ c22̄ , e2ξ2+0 = c22̄. Acknowledgements We greatly appreciate all referees’ useful comments which help us improve the present paper significantly. The work of B.F. is partially supported by NSF Grant (No. 1715991) and the COS Research Enhancement Seed Grants Program at UTRGV. 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N -Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation

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