$N$-Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schr\"odinger 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-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.

The integrable discretization of nonlinear Schrödinger equation iq n,t = 1 + σ|q n | 2 (q n+1 + q n−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 The coupled nonlinear Schrödinger equation iu t = u xx + 2 σ 1 |u| 2 + σ 2 |v| 2 u, iv t = v xx + 2 σ 1 |u| 2 + σ 2 |v| 2 v, (1.1) where σ i = ±1, i = 1, 2, was firstly recognized being integrable by Yajima and Oikawa [47]. For the focusing-focusing case (σ 1 = σ 2 = 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 equation, the N -bright soliton solution was obtained in [16,22,51]; the general bright-dark and darkdark 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 where σ i = ±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 (σ 1 = σ 2 = 1), and is of dark type for the defocusing-defocusing case (σ 1 = σ 2 = −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) where σ µ = ±1 for µ = 1, . . . M . For all-focusing case (σ µ = 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 σ µ ) is missing. The aim of the present paper is to construct a general N -bright-dark soliton solution to the semidiscrete 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 where j = 1, . . . , m, ω l = s(a l −ā l ), |a l | = 1 withā l representing the complex conjugate of a l , l = 1, . . . , M − m. Here, f n is a real-valued function, whereas, g n and h n are complex-valued functions. The transformations convert equation (1.3) into a set of bilinear equations as follows n−1 f n+1 , j = 1, . . . , m, In what follows, we give a Pfaffian-type solution to the above bilinear equations.
Proof . It can be shown that , so as other similar Pfaffians. Thus, an algebraic identity of Pfaffian together with above Pfaffian relations gives which is exactly the first bilinear equation. Next we prove the second bilinear equation. It can also be shown that Therefore, an algebraic identity of Pfaffian together with above Pfaffian relations gives 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 Then from the fact we then have Then we can show On the other hand, since Similarly, we can show Finally, we have 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 twoand three-coupled discrete NLS equation

Two-component semi-discrete NLS equation
In this subsection, we provide and illustrate one-and two-soliton for two-component semidiscrete NLS equation (1.2) explicitly.
One-soliton solution. the tau-functions for one-soliton solution (N = 1) are The above tau functions lead to the one-soliton solution as follows a 1 p 1 )(1 − a 1p1 )). Therefore, the amplitude of bright soliton for q (1) are 1 2 α 1 / √ a 11 . The dark soliton q (2) approaches |ρ 1 | as x → ±∞. In addition, the intensity of the dark soliton is |ρ 1 | cos φ 1 . An example of one-bright-dark soliton is illustrated in Fig. 1a   Two-soliton solution. The tau functions for two-soliton are of the following form A two bright-dark soliton solution is shown in Fig. 2 before and after the collision for parameters σ 1 = 1.0, σ 2 = −1, p 1 = 1.0 + 0.5i, ρ = 2.0, α 2 = 1.0 + 0.5i, a 1 = 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.

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 Let u n = x n + iy n and v n = z n + iw n , then equation (4.1) becomes d dt x n = 1 + x 2 n + y 2 n + z 2 n + w 2 n (y n+1 + y n−1 ), − d dt y n = 1 + x 2 n + y 2 n + z 2 n + w 2 n (x n+1 + x n−1 ), d dt z n = 1 + x 2 n + y 2 n + z 2 n + w 2 n (w n+1 + w n−1 ), − d dt w n = 1 + x 2 n + y 2 n + z 2 n + w 2 n (z n+1 + z n−1 ).

By defining
x n , for n even, y n , for n odd, U (2) n = −y n , for n even, x n , for n odd, z n , for n even, w n , for n odd, U (4) n = −w n , for n even, z n , for n odd, U (5) n = (−1) n/2 , for n even, (−1) (n−1)/2 , for n odd, we obtain for n being even, and for n being odd, in other words, Consequently, n , n even, U n , and v n and v * n correspond to U n and U 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 Second, we give a comparison between the NLS-type equations and the sine/sinh-Gordontype 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], a generic integrable relativistic system associated with the sl(2, C) was systematically investigated, 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 exhibits nontrivial interaction of solitons such as decay and fusion [10]. Here u 1 and u 2 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 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 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 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, 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(a 1 , a 2 , a 3 , a 4 , b 1 , b 2 , b 3