Gram Determinant Solutions to Nonlocal Integrable Discrete Nonlinear Schrodinger¨ Equations Via the Pair Reduction

Gram determinant solutions to nonlocal integrable discrete nonlinear Schrodinger¨ equations via the pair reduction

Junchao Chen a,∗, Bao-Feng Feng b, Yongyang Jin c

aDepartment of Mathematics and Institute of Nonlinear Analysis, Lishui University, Lishui, 323000, China bSchool of Mathematical and Statistical Sciences, The University of Texas-Rio Grande Valley, Edinburg, TX 78541, USA cDepartment of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310023, China

Abstract

In this paper, Gram determinant solutions of local and nonlocal integrable discrete nonlinear Schrodinger¨ (IDNLS) equations are studied via the pair reduction. A generalized IDNLS equation is ﬁrstly introduced which possesses the single Casorati determinant solution. Two kinds of Gram determinant solutions are presented from Casorati determinant ones due to the gauge freedom. The different pair constraint conditions for wave numbers are imposed and then solutions of local and nonlocal IDNLS equations are derived in terms of Gram determinant. Keywords: nonlocal discrete NLS equation, bilinear form, pair reduction, solution, Gram determinant

1. Introduction

The parity-time (PT) symmetry has firstly proposed in quantum mechanics since Bender and Boettcher [1] found that non-Hermitian Hamiltonians possess entirely real spectra. In nonlinear integrable system, Ablowitz and Musslimani have recently introduced a nonlocal nonlinear Schrodinger¨ (NLS) equation with the PT-symmetric invariance. Such a nonlocal PT-symmetric NLS equation remains integrable due to the existence of a Lax pair formulation and an infinite number of conservation laws. Indeed, it can be obtained from a nonlocal reduction of the AKNS spectral problem [2, 3]. Along with this idea, many nonlocal versions of local integrable equations have been identified in both one and two space dimensions as well as in discrete case [2–11]. These nonlocal reductions include the reverse space-time symmetry, or the partially PT symmetry and the partially reverse space- time symmetry in higher dimensional case. The potential applications of nonlocal soliton equations appear in nonlinear PT symmetric media [6], or more universally in the context of “Alice-Bob events” [7, 8]. Moreover, the classical methods such as inverse scattering transform, Darboux transformation and bilinear approach have recently applied to nonlocal integrable models and gave rise to new types of solution [12–27]. Among these nonlocal system, three kinds of nonlocal integrable discrete NLS (IDNLS) equations can be reduced from the Ablowitz-Ladik (AL) spectral problem [3]. More specifically, considering the AL scattering problem        i −1 2 −1   z ψ  iψ ϕ − − (z − z ) −i(zψ − z ψ − )   n   n n 1 2 n n 1  υn+1 =   υn, υn,t =   υn, (1) ϕ −1 −1ϕ − ϕ − ϕ ψ + i − −1 2 n z i(z n z n−1) i n n−1 2 (z z )

(1) (2) T where υn = (υn , υn ) and z is a complex spectral parameter, the discrete compatibility condition υn+1,t =

(υm,t)m=n+1 yields the coupled system

iψn,t = ψn+1 + ψn−1 − 2ψn − ψnϕn(ψn+1 + ψn−1), (2)

−iϕn,t = ϕn+1 + ϕn−1 − 2ϕn − ϕnψn(ϕn+1 + ϕn−1). (3)

∗Corresponding author. Department of Mathematics, Lishui University, Lishui, 323000, China Email address: [email protected] ( Junchao Chen ) Preprint submitted to Elsevier November 23, 2019 It admits four different symmetry reductions [3, 12, 13, 19, 22, 24]: ϕ = δψ∗ (i) The standard Ablowitz-Ladik symmetry ( n n) gives rise to the local IDNLS equation

ψ = ψ + ψ − ψ − δψ ψ∗ ψ + ψ , δ = ± . i n,t n+1 n−1 2 n n n( n+1 n−1) 1 (4)

ϕ = δψ∗ (ii) The discrete PT preserved symmetry ( n −n) leads to the PT symmetric IDNLS equation

ψ = ψ + ψ − ψ − δψ ψ∗ ψ + ψ , δ = ± . i n,t n+1 n−1 2 n n −n( n+1 n−1) 1 (5)

(iii) The reverse time symmetry (ϕn = γψn(−t)) yields the reverse time symmetric IDNLS equation

iψn,t = ψn+1 + ψn−1 − 2ψn − γψnψn(−t)(ψn+1 + ψn−1), (6) where γ is an arbitrary complex constant.

(iv) The reverse discrete-time symmetry (ϕn = γψ−n(−t)) results in the reverse discrete-time symmetric IDNLS equation

iψn,t = ψn+1 + ψn−1 − 2ψn − γψnψ−n(−t)(ψn+1 + ψn−1), (7) where γ is an arbitrary complex constant. More recently, a bilinearisation-reduction approach [24] has been proposed to derive solutions of Eqs.(4)-(7). By imposing different reduction conditions, double Casoratian solutions for Eqs.(4)-(7) have been provided [24]. It is known that the local IDNLS equation (4) admits bright solition solutions in terms of the double Casorati determinant but dark soliton ones expressed as the single Casorati determinant [28]. Thus, “dark” soliton solutions for nonlocal IDNLS equations (5)-(7) need to be investigated. Apart from the single Casorati determinant form, the dark soliton solution of the local IDNLS equation (4) can be written as Gram determinant form. It motivates us to construct Gram determinant solutions for nonlocal IDNLS equations. In the previous work [27], one of authors have used the direct reduction, namely the wave number satisfying the constraint condition under the same index, to obtain Gram determinant solutions for local and nonlocal IDNLS equations (4)-(6). For instance, the derivation of the dark soliton solution for the defocusing local IDNLS Eq.(4) was realized by imposing the constraint condition = ∗−1 qi pi [27, 28] on the before-redcution IDNLS equation. In the direct reduction, only singular soliton solutions were obtained for the nonlocal PT symmetric IDNLS Eq.(5) and only one-soliton solution was derived for the reverse time symmetric IDNLS Eq.(6)[27]. Note that in the continuous nonlocal PT-symmetric NLS equation with the nonzero boundary condition case, there existed an even number of soliton solutions related to an even number (2N) of eigenvalues [4, 26]. So it is necessary to develop a kind of pair reduction on the before-redcution IDNLS equation, in which the constraint condition is taken between a pair of wave numbers, to derive dark solutions for local and nonlocal IDNLS equations. However, for the reverse discrete-time symmetric IDNLS Eq.(7), the soliton solution can be obtained directly from the one of the before-redcution IDNLS equation without the constraint condition for wave numbers. Therefore, in the present paper, we will apply the pair reduction to construct Gram determinant solutions for the local Eq.(4), the PT symmetric Eq.(5) and the reverse time symmetric Eq.(6). This paper is organized as follows. In Section 2, we ﬁrst introduce a generalized (before-redcution) IDNLS equation whose solution is expressed in terms of the single Casorati determinant. Then we convert the Casorati determinant solution to two kinds of the Gram determinant one. In Section 3, with the help of the pair reduction, Gram determinant solutions of local and nonlocal IDNLS equations are derived by imposing different constraint conditions. The last section is a summary and discussion.

2 2. Gram determinant solution of the generalized IDNLS equation

2.1. Casorati determinant solution In order to derive Gram determinant solution from the known Casorati determinant one for the generalized IDNLS equation, we ﬁrst recall the derivation of Casorati determinant solution in [28, 29]. The bilinear forms for Backlund¨ Transformation (BT) of Toda lattice (TL) equation

(aDx − 1)τn+1(k + 1, l) · τn(k, l) + τn(k + 1, l)τn+1(k, l) = 0, (8)

(bDy − 1)τn−1(k, l + 1) · τn(k, l) + τn(k, l + 1)τn−1(k, l) = 0, (9) and the bilinear form of discrete 2-dimensional Toda lattice (D2DTL) equation

τn(k + 1, l + 1)τn(k, l) − τn(k + 1, l)τn(k, l + 1) = ab[τn(k + 1, l + 1)τn(k, l) − τn+1(k + 1, l)τn−1(k, l + 1)], (10) with the constants a and b, have the following Casorati determinant solution

+ + − φ(n)(k, l) φ(n 1)(k, l) ··· φ(n N 1)(k, l) 1 1 1 + + − φ(n)(k, l) φ(n 1)(k, l) ··· φ(n N 1)(k, l) τ , = 2 2 2 , n(k l) . . . (11) . . ··· .

φ(n) , φ(n+1) , ··· φ(n+N−1) , N (k l) N (k l) N (k l)

φ(n) , where i (k l) are functions of continuous independent variables x, y and discrete ones k, l, satisfying the dispersion relations as follows

∂ φ(n) , = φ(n+1) , , ∂ φ(n) , = φ(n−1) , , x i (k l) i (k l) y i (k l) i (k l) (12) ∆ φ(n) , = φ(n+1) , , ∆ φ(n) , = φ(n−1) , . k i (k l) i (k l) l i (k l) i (k l) (13)

Here ∆k and ∆l are the backward difference operators with respect to the difference intervals a and b given by

f (k, l) − f (k − 1, l) f (k, l) − f (k, l − 1) ∆ f (k, l) = , ∆ f (k, l) = , (14) k a l a and the Hirota’s bilinear operators Dx and Dy are deﬁned as ( ) ( ) ∂ ∂ n ∂ ∂ m n m · = − − , ′, ′ . DxDy (a b) ′ ′ a(x y)b(x y ) ∂x ∂x ∂y ∂y x=x′,y=y′

The Casorati determinant solution (11) can give rise to various types of solutions for the bilinear equations (8)-(10), since the matrix elements can be taken as any functions obeying the dispersion relations (12)-(13). In the φ(n) , following, the function i (k l) is taken as ( ) ( ) −l −l φ(n) , = n − −k − 1 ξ + n − −k − 1 η , i (k l) pi (1 api) 1 b exp( i) qi (1 aqi) 1 b exp( i) (15) pi qi with 1 1 ξi = pi x + y + ξi,0, ηi = qi x + y + ηi,0, pi qi where pi, qi and ξi,0, ηi,0 are arbitrary constants. This kind of choice leads to soliton solutions and pi, qi and ξi,0,

ηi,0 represent the wave numbers and phase parameters of solitons, respectively. In order to reduce the coupled system of TL’s BT (8)-(9) and D2DTL (10) to the generalized IDNLS equation,

3 one need to impose the constraint condition for the wave numbers in (15)

apiqi + b − pi − qi = 0, (16) which implies 1 − b 1 1 − b 1 2 pi = 2 qi . pi qi (17) 1 − api 1 − aqi φ(n) , This condition makes i (k l) in (15) satisfy

1 − b 1 φ(n+2) + , − = 2 pi φ(n) , , i (k 1 l 1) pi i (k l) (18) 1 − api and then tau functions have the relations   ∏N 1 − b 1   2 pi  τn+2(k + 1, l − 1) =  p  τn(k, l). (19) i 1 − ap i=1 i

Therefore, the bilinear forms of TL’s BT (8)-(9) and D2DTL (10) become

(aDx − 1)τn+1(k + 1, l) · τn(k, l) + τn(k + 1, l)τn+1(k, l) = 0, (20)

(bDy − 1)τn+1(k + 1, l) · τn(k, l) + τn+2(k + 1, l)τn−1(k, l) = 0, (21)

τn+1(k + 1, l)τn−1(k − 1, l) − τn+1(k, l)τn−1(k, l) = ab[τn+1(k + 1, l)τn−1(k − 1, l) − τn(k, l)τn(k, l)]. (22)

Here the parameter l is dropped by simply taking l = 0. Furthermore, by using the independent variable transformation

x = iact, y = ibdt, i.e., − i∂t = ac∂x + bd∂y, (23) where c and d are constants, and deﬁning

fn = τn(0), gn = τn+1(1), hn = τn−1(−1), (24) the above bilinear equations converts to the bilinear form of the generalized IDNLS equation

(−iDt − c − d)gn · fn + dgn+1 fn−1 + cgn−1 fn+1 = 0, (25)

(iDt − c − d)hn · fn + chn+1 fn−1 + dhn−1 fn+1 = 0, (26) − 2 = − 2 − , fn+1 fn−1 fn (ab 1)( fn gnhn) (27) which have the solution in terms of Casorati determinant as follows

+ − + − n j 1 ξi n j 1 ηi fn = |F| = p e + q e , (28) i i n+ j n+ j p ξ q η i i i i gn = |G| = e + e , (29) 1 − ap 1 − aq i i + − ξ + − η = | | = − n j 2 i + − n j 2 i , hn H (1 api)pi e (1 aqi)qi e (30) with ( ) ( ) bd bd ξi = i acpi + t + ξi,0, ηi = i acqi + t + ηi,0 (31) pi qi where the wave numbers pi and qi need to satisfy the constraint condition (16).

4 Through the dependent variable transformation

gn hn un = , vn = (32) fn fn one can get the generalized IDNLS equation

dun i + (c + d)u − [ab − (ab − 1)u v ](du + + cu − ) = 0, (33) dt n n n n 1 n 1 dvn −i + (c + d)v − [ab − (ab − 1)u v ](cv + + dv − ) = 0. (34) dt n n n n 1 n 1

2.2. Gram determinant solution In this section, we will transform the above Casorati determinant solution to two kinds of Gram determinant one for the generalized IDNLS equation. To this end, we change exponential functions as      −1  −1 ∏N  ∏N    ′   ′ exp(ξ ) =  (q − p ) exp(ξ ), exp(η ) =  (q − q ) exp(η ), (35) i  k i  i i  k i  i k=1 k=1 k,i k,i with ( ) ( ) ξ′ = + bd + ξ′ , η′ = + bd + η′ , i i acpi t i,0 i i acqi t i,0 pi qi and introduce the following 2N × 2N diagonal matrices

1 A = Diag(a1, a2 ··· , a2N ), ai = , (36) qi − pi = , · ·· , , = 1 , B Diag(b1 b2 b2N ) bi ξ′ (37) n i pi e 1 − api C = Diag(c1, c2, · ·· , c2N ), ci = , (38) pi and a Vandermonde matrix       −   ∑ 2∏N i  =  − 2N−i   . Vq ( 1)  qkl  (39)   1≤k1

Due to the gauge freedom, we can ﬁnd the ﬁrst kind of Gram determinant solution

n η′ q e i ˜ = | ˜| = | | = 1 + δ 1 i , fn F ABFVq i j ξ′ (42) pi − q j pi − q j pne j j 2N×2N [ ] ′ n η − q e i = | ˜| = | | = 1 + δ qi(1 api) 1 i , g˜n G CABFVq i j ξ′ (43) pi − q j p j(1 − aq j) pi − q j pne j j 2N ×2N [ ] ′ n η − q e i ˜ = | ˜ | = | −1 | = 1 + δ pi(1 aqi) 1 i , hn H C ABFVq i j ξ′ (44) pi − q j q j(1 − ap j) pi − q j pne j j 2N×2N 5 still satisfy the bilinear IDNLS equations (25)-(27). For the second case, we need to introduce another group of 2N × 2N diagonal matrices

P = Diag(1, 1 ··· , 1N ; pN+1, pN+2, ··· , p2N ), (45)

Q = Diag(1, 1 ··· , 1N ; qN+1, qN+2, ··· , q2N ). (46)

Owing to the gauge freedom, one can derive the second kind of Gram determinant solution

AB f˜ = |F˜| = |PABFV Q| = , (47) n q CD 2N×2N

A B g˜ = |G˜| = |CPABFV Q| = (g) , (48) n q CD(g) 2N×2N

A B h˜ = |H˜ | = |C−1PABFV Q| = (h) , (49) n q CD(h) 2N×2N with the block matrices given by ( ) ( ) ( ) A = ai j(0) , A g = ai j(1) , A h = ai j(−1) , (50) ( )N×N ( ) ( )N×N ( ) ( )N×N D = di j(0) , D g = di j(1) , D h = di j(−1) , (51) ( ) N×N (( ) ) N×N ( ) N×N B = bi j , C = ci j , (52) N×N N×N where the elements are deﬁned by [ ] k n η′ − q e i q + = 1 + δ 1 qi(1 api) i , = N j , ai j(k) i j ξ′ bi j (53) pi − q j pi − q j p j(1 − aq j) n j pi − qN+ j p j e [ ] k n η′ p + q + p + q + − q e N+i = pN+i , = N i N j + δ N i N j qN+i(1 apN+i) N+i . ci j di j(k) i j ξ′ (54) pN+i − q j pN+i − qN+ j pN+i − qN+ j pN+ j(1 − aqN+ j) n N+ j pN+ je

This Gram determinant solution also satisfy the bilinear IDNLS equations (25)-(27).

3. Reduction to local and nonlocal IDNLS equations

3.1. The local IDNLS equation (4) For the local IDNLS equation (4), we start from the second kind of Gram determinant solution (47)-(49) via the pair reduction. To be speciﬁc, by imposing the pair conditions

1 1 ∗ pN+i = ∗ , qN+i = ∗ , b = a , (55) qi pi one can ﬁnd that the constraint condition (16) for pN+i and qN+i is complex conjugate of one for pi and qi. If we further restrict = ∗, ξ′ = −η′∗ , η′ = −ξ′∗ , d c N+i,0 i,0 N+i,0 i,0 (56) then the following relations hold

′ ′∗ ′ ′∗ ξ′ ∗− −η′∗ η′ ∗− −ξ′∗ ξ = −η , η = −ξ , n N+i = n i , n N+i = n i . N+i i N+i i pN+ie qi e qN+ie pi e (57)

6 The elements in the Gram determinant solution (47)-(49) are rewritten as [ ] k n η′ − q e i = 1 + δ 1 qi(1 api) i , = 1 , ai j(k) i j ξ′ bi j ∗ (58) pi − q j pi − q j pi(1 − aqi) n j pi p − 1 p j e j [ ] ∗ η′∗ − ∗ k q ne j 1 1 1 a qi j c = , d k = + δ ′∗ . i j ∗ i j( ) ∗ ∗ i j ∗ ∗ ∗ ∗n ξ (59) − q q p − q p − q a − p i 1 i j j i j i i pi e

∗ T ∗ T ∗ T ∗ ∗ In this case, we have A = D , B = B and C = C . Notice that a − pi = qi(1 − api) and a − qi = pi(1 − aqi) A∗ = DT A∗ = DT which yield (g) (h) and (h) (g), thus one can get

˜∗ = | ˜T | = | ˜| = ˜ , ∗ = | ˜ T | = | ˜| = ˜ , fn T F T F fn gñ T H T F hn (60) where the matrix T is defined by      0 IN×N  T =   . (61) IN×N 0 2N×2N ∗ ˜ ∗ = gñ = hn = From the transformation (32), it immediately reaches un ∗ vn. Furthermore, through the variable fñ fñ transformations ( ) ψ eiθ + e−iθ |a|2 − 1 e−iθ u = √n exp −inθ + i t − 2it , α = , c = , (62) n α |a|2 δ|a|2 |a|2 with δ = 1(|a|2 > 1) and δ = −1(|a|2 < 1), the generalized IDNLS equation (33)-(34) reduces to the local IDNLS equation (4). Finally, we arrive at the following theorem about 2N-soliton solution of the local IDNLS equation (4). Theorem 3.1 The local IDNLS equation (4) has the solution √ ( ) 2 iθ+ −iθ |a| − 1 inθ−i e e t+2it gñ ψ = e |a|2 , (63) n δ| |2 a fñ with δ = 1(|a|2 > 1) and δ = −1(|a|2 < 1), where tau functions are given by

AB A B f˜ = , g˜ = (g) , (64) n n CD CD(g) 2N×2N 2N×2N with the block matrices    [ ]   n η′   n η′   q e i   − q e i  A =  1 + δ 1 i  , A =  1 + δ 1 qi(1 api) i  ,  i j ξ′  (g)  i j ξ′  pi − q j pi − q j pne j pi − q j pi − q j pi(1 − aqi) pne j  j N×N  j N×N ∗ η′∗ [ ] ∗ η′∗  n j   ∗ n j   1 1 q e   1 1 a − q q e  D =  + δ j  , D =  + δ i j  ,  ∗ ∗ i j ∗ ∗ ∗n ξ′∗  (g)  ∗ ∗ i j ∗ ∗ ∗ ∗n ξ′∗  p − q p − q p e i p − q p − q a − p p e i j i j i i N×N j i j i i i N×N and   ( )    1  1 B =  ∗  , C = ∗ . pi p − 1 1 − q q j j N×N i N×N ( ) ( ) ξ′ = ae−iθ + a∗eiθ + ξ′ η′ = ae−iθ + a∗eiθ + η′ ξ′ η′ = , · ·· , Here i 2 pi 2 t and i 2 qi 2 t , pi, qi , (i 1 2 N) and a are arbitrary i |a| |a| pi i,0 i |a| |a| qi i,0 i,0 i,0 complex constants, θ is an arbitrary real constant, these parameters need to satisfy the constraint conditions

∗ apiqi + a − pi − qi = 0. (65)

The above solution holds for both focusing and defocusing cases in the local IDNLS equation (4). As pointed out in [28], this kind of solution corresponds to the homoclinic orbit one of the local IDNLS equation. 7 For example, by taking N = 1, tau functions for the local IDNLS equation (4) are written as:

 ′ ′∗  n η ∗n η 2n η′ +η′∗  q e 1 q e 1 | | 1 1  = ∆ + ∆  + 1 + 1 + q1 e  , fn 1 2 1 ξ′ ∗ ξ′∗ ξ′ +ξ′∗  (66) n 1 n 1 | |2n 1 1 p1e p1 e p1 e  ′ ′∗  n η ∗n η 2n η′ +η′∗  q e 1 q e 1 | | 1 1  = ∆ + ∆  + 1 + ∗ 1 + | |2 q1 e  , gn 1 2 1 P1 ξ′ P1 ∗ ξ′∗ P1 ξ′ +ξ′∗  (67) n 1 n 1 | |2n 1 1 p1e p1 e p1 e where 1 1 a∗ − p ∆ = , ∆ = , P = 1 , 1 2 2 2 2 1 ∗ (|p1| − 1)(|q1| − 1) |p1 − q1| a − q1 ( ) ( ) ξ′ = ae−iθ + a∗eiθ + ξ′ η′ = ae−iθ + a∗eiθ + η′ ξ′ η′ and i 2 p1 2 t and i 2 q1 2 t , p1, q1 , and a are arbitrary complex 1 |a| |a| p1 1,0 1 |a| |a| q1 1,0 1,0 1,0 ∗ constants, θ is an arbitrary real constant, they need to satisfy the constraint condition ap1q1 + a − p1 − q1 = 0.

3.2. The PT symmetric IDNLS equation (5) For nonlocal IDNLS equation (5), we carry out the pair reduction on the ﬁrst kind of Gram determinant solution (42)-(44). For this purpose, we ﬁrst impose the pair conditions

= ∗, = ∗, ∗ = , ∗ = , pN+i qi qN+i pi a a b b (68) which ensure that the constraint condition (16) for pN+i and qN+i is complex conjugate of one for pi and qi. If we further require ∗ = , ∗ = , ξ′ = −η′∗ , η′ = −ξ′∗ , c c d d N+i,0 i,0 N+i,0 i,0 (69) one can ﬁnd the relations

′ ′∗ ′ ′∗ ξ′ ∗ −η′∗ η′ ∗ −ξ′∗ ξ = −η , η = −ξ , n N+i = n i , n N+i = n i . N+i i N+i i pN+ie qi e qN+ie pi e (70)

The Gram determinant solution (42)-(44) can be rewritten as

AB A B A B f˜ = |F˜| = , g˜ = |G˜| = (g) , h˜ = |H˜ | = (h) , (71) n n n CD CD(g) CD(h) 2N×2N 2N×2N 2N×2N with the block matrices given by (50)-(52) whose elements are deﬁned by [ ] k n η′ − q e i = 1 + δ 1 qi(1 api) i , = 1 , ai j(k) i j ξ′ bi j ∗ (72) pi − q j pi − q j p j(1 − aq j) n j pi − p p j e j [ ] ∗ ∗ k ∗n η′∗ p (1 − aq ) p e i = 1 , = 1 + δ 1 i i i . ci j ∗ di j(k) ∗ ∗ i j ∗ ∗ ∗ ∗ ∗ ξ′∗ (73) q − q j q − p q − p q (1 − ap ) n j i i j i j i i q j e

A∗ − = −DT A∗ − = −DT A∗ − = −DT B∗ = −BT C∗ = −CT In this situation, we have ( n) , ( n)(g) (h), ( n)(h) (g), and , thus one can get ˜∗ = | − ˜T | = | ˜| = ˜ , ∗ = | − ˜ T | = | ˜| = ˜ . f−n T F T F fn g˜−n T H T F hn (74)

∗ ˜ ∗ = g˜−n = hn = From the transformation (32), it immediately reaches u−n ˜∗ ˜ vn. Furthermore, through the variable f−n fn transformations ( ) ψ eθ + e−θ ab − 1 e−θ eθ u = √n exp −nθ + i t − 2it , α = , c = , d = , (75) n α ab δab ab ab with δ = 1(ab > 1, or ab < 0) and δ = −1(0 < ab < 1), we obtain the PT symmetric IDNLS equation (5). Finally, we get the following theorem about the solution of the nonlocal IDNLS equation (5).

8 Theorem 3.2 The nonlocal PT symmetric IDNLS equation (5) has the solution

√ ( ) θ −θ ab − 1 nθ−i e +e t+2it g˜ ψ = ab n , n δ e (76) ab f˜n with δ = 1(ab > 1, or ab < 0) and δ = −1(0 < ab < 1), where tau functions are given by

AB A B f˜ = , g˜ = (g) , (77) n n CD CD(g) 2N×2N 2N×2N with the block matrices    [ ]   n η′   n η′   q e i   − q e i  A =  1 + δ 1 i  , A =  1 + δ 1 qi(1 api) i  ,  i j ξ′  (g)  i j ξ′  pi − q j pi − q j n j pi − q j pi − q j p j(1 − aq j) n j p j e × p j e ×  N N  [ ] N N  ∗n η′∗   ∗ ∗ ∗n η′∗   p e i   p (1 − aq ) p e i   1 1 i   1 1 i i i  D =  + δi j ′∗  , D(g) =  + δi j ′∗  ,  ∗ − ∗ ∗ − ∗ ∗n ξ   ∗ − ∗ ∗ − ∗ ∗ − ∗ ∗n ξ  qi p j qi p j q e j qi p j qi p j qi (1 api ) q e j j N×N j N×N and   ( )    1  1 B =  ∗  , C = ∗ . pi − p q − q j j N×N i N×N ( ) ( ) ′ e−θ eθ ′ ′ e−θ eθ ′ ′ ′ Here ξ = i pi + t + ξ and η = i qi + t + η , pi, qi ξ and η (i = 1, 2 · ·· , N) are arbitrary i b api i,0 i b aqi i,0 i,0 i,0 complex constants, a, b and θ are arbitrary real constants, these parameters need to satisfy the constraint conditions

apiqi + b − pi − qi = 0. (78)

For example, tau functions for the nonlocal IDNLS equation (5) are written as:   n η′ ∗n η′∗ n ∗n η′ +η′∗  q e 1 p e 1 q p e 1 1   1 1 1 1  f = ∆ + ∆  + ′ + ′∗ + ′ ′∗  , n 1 2 1 n ξ ∗n ξ n ∗n ξ +ξ (79) p e 1 q e 1 p q e 1 1  1 1 1 1  n η′ ∗n η′∗ n ∗n η′ +η′∗  q e 1 p e 1 q p e 1 1   1 1 1 P1 1 1  g = ∆ + ∆  + P ′ + ′∗ + ′ ′∗  n 1 2 1 1 n ξ ∗ ∗n ξ ∗ n ∗n ξ +ξ (80) 1 P 1 P 1 1 p1e 1 q1 e 1 p1q1 e where 1 1 q (1 − ap ) ∆ = , ∆ = − , P = 1 1 , 1 − ∗ − ∗ 2 | − |2 1 − (p1 p1)(q1 q1) p1 q1 p1(1 aq1) ( ) ( ) ′ e−θ eθ ′ ′ e−θ eθ ′ ′ ′ and ξ = i p1 + t + ξ and η = i q1 + t + η , p1, q1 ξ and η are arbitrary complex constants, 1 b ap1 1,0 1 b aq1 1,0 1,0 1,0 a, b and θ are arbitrary real constants, they need to satisfy the constraint condition ap1q1 + b − p1 − q1 = 0. To investigate the asymptotic behavior of above two-soliton solution (79)-(80), we redeﬁne the following notations: √ eθ + e−θ ab − 1 q ∆ 1 ρ+iφ c0+iθ1 2 c1 θˆ = nθ − i t + 2it, ρˆ = , = e , P1 = e , = δˆe , (81) ab δab p1 ∆1 + ∆2 n η′ ∗n η′∗ ′ ′ q e 1 χ p e 1 χ η − ξ = + + + , 1 = 1 , 1 = 2 . 1 1 (A iB)t A0 iB0 ξ′ e ∗ ξ′∗ e (82) n 1 n 1 p1e q1 e

Without loss of generality, we assume A > 0 and then have the asymptotic forms as follows: (i) Before collision (t → −∞)

Soliton 1 (χ1 ≈ 0, χ2 ≈ −∞):

+ + + + X +c1+c0 i(Y +θ1) + + + δˆ + θ θˆ 1 + δˆe 1 e 1 ab − 1 θ+ cosh(X c1 c0) cos(Y 1) ψ → ρ , |ψ |2 → 2n c0 1 1 . n ˆe ++ + n e + + (83) + δˆ X1 c1 iY1 δab + + δˆ 1 e e cosh(X1 c1) cos(Y1 )

9 Soliton 2 (χ2 ≈ 0, χ1 ≈ −∞):

− − − − −X +c1−c0 i(Y +θ1) − + + δˆ + θ θˆ 1 + δˆe 1 e 1 ab − 1 θ− cosh(X c1 c0) cos(Y 1) ψ → ρ , |ψ |2 → 2n c0 1 1 . n ˆe − −+ − n e − − (84) + δˆ X1 c1 iY1 δab − + δˆ 1 e e cosh(X1 c1) cos(Y1 )

(ii) After collision (t → +∞)

Soliton 1 (χ1 ≈ 0, χ2 ≈ +∞):

+ + + − −X −c0 −i(Y +θ1) + + + θ θˆ p 1 + e 1 e 1 ab − 1 θ− cosh(X c0) cos(Y 1) 1 2 2n c0 1 1 ψ → ρˆe + + , |ψ | → e . (85) n ∗ −X − Y n + + p + 1 i 1 δab + δˆ 1 1 e e cosh(X1 ) cos(Y1 )

Soliton 2 (χ2 ≈ 0, χ1 ≈ +∞):

− − − − X +c0 −i(Y +θ1) + + + θ θˆ p 1 + e 1 e 1 ab − 1 θ+ cosh(X c0) cos(Y 1) 1 2 2n c0 1 1 ψ → ρˆe − − , |ψ | → e . (86) n ∗ X − Y n − − p + 1 i 1 δab + δˆ 1 1 e e cosh(X1 ) cos(Y1 ) ± = ρ ± ± ± = φ ± ± Here X1 n At A0 and Y1 n Bt B0.

(a) (b)

= = 18 = + = 3 + 3 Figure 1: The solution (79)-(80) of the PT symmetric IDNLS equation (5) with the parameters (a) a 1, b 5 , p1 3 3i and q1 5 5 i; = = = + = + = = = + = = = 3 = + = 19 + 3 (b) a 1, b 3, p1 1 i and q1 1 2i; (c) a 1, b 11, p1 2 3i and q1 3i; (d) a 1, b 2 , p1 2 3i and q1 20 20 i.

= ∗ In the previous work [27], one of authors have applied a direct reduction qi pi to derive the solution of the nonlocal IDNLS equation (5). When N = 1, the solution reads ( ) − ∗ n ′ ′∗ √ ( ) [ ] p1(1 ap1) p1 ξ +ξ θ −θ ∗ + ∗ ∗ 1 1 e +e − 1 − e ab − 1 nθ−i t+2it p (1 ap1) p1(1 ap1) p1 ψ = ab 1 ( ) , n e ∗ n (87) δ − ′ ′∗ ab p1(1 ap ) p1 ξ +ξ 1 1 + ∗ e 1 1 p1 ( ) θ ′ pi e ′ ′ where, δ = 1(ab < 0) and δ = −1(0 < ab < 1), ξ = i θ + t + ξ , p1 and ξ are arbitrary complex i be api i,0 1,0 , θ − = − p1 − b < parameters, a b and are arbitrary real parameters and they need to satisfy 1 ap1 p∗ (1 p ) and ab 1. For [ ] 1 1 (1−ap∗) 1 2iγ1 ∗ the solution (87), if we set pi = exp(αi + iβi), = e and deﬁne ξ1 + ξ ≡ 2ϑ1, then the modular square of (1−ap1) 1 10 ψ n is given by [ ] − β + β + γ β + γ 2 1 ab 2 sin(2n 1 1 ) sin( 1 ) |ψn| = exp(2nθ) 1 − , (88) ab cosh(2ϑ1) + cos(2nβ1) which means the singularity at β1 = kπ (k = ±1, ±2, ··· ). However, the solution (79)-(80) may allow the nonsingular case under the suitable choice of parameters. From the above asymptotic analysis, it is found that the requirement φ = B = B0 = 0 is an sufﬁcient condition for ξ′ = η′ = the nonsingular solution. We illustrate the solution (79)-(80) in Fig.1 with the parameters 1,0 1,0 0, in which θ is taken as zero to avoid the amplitude increases/decreases exponentially. Under the special parameters satisfying φ = B = B0 = 0, Figure 1(a) exhibits a nonsingular two-dark soliton. Other three examples in Fig.1 are singular solutions, although they look like two kink soliton, the interaction of kink and breathing soliton and two breathing-soliton, respectively.

3.3. The reverse time discrete symmetric IDNLS equation (6) For the nonlocal IDNLS equation (6), we consider the pair reduction on the second kind of Gram determinant solution (47)-(49). More speciﬁcly, by imposing the pair conditions

1 1 pN+i = , qN+i = , b = a, (89) qi pi it is found that the constraint condition (16) for pN+i and qN+i is consistent with one for pi and qi. If we further require = , ξ′ = −η′ , η′ = −ξ′ , d c N+i,0 i,0 N+i,0 i,0 (90) then the following relations hold

′ ′ ′ ′ ξ′ − −η′ − η′ − −ξ′ − ξ = −η − , η = −ξ − , n N+i = n i ( t), n N+i = n i ( t). N+i i ( t) N+i i ( t) pN+ie qi e qN+ie pi e (91)

The elements in the Gram determinant solution (47)-(49) are rewritten as [ ] k n η′ − q e i = 1 + δ 1 qi(1 api) i , = 1 , ai j(k) i j ξ′ bi j (92) pi − q j pi − q j pi(1 − aqi) n j pi p j − 1 p j e [ ] η′ − k n j( t) 1 1 1 a − qi q j e c = , d (k) = + δ ′ . (93) i j i j i j n ξ (−t) − q q p − q p − q a − p i 1 i j j i j i i pi e

T T T In this case, we bave A(−t) = D , B = B and C = C . Notice that a − pi = qi(1 − api) and a − qi = pi(1 − aqi) A − = DT A − = DT which yield (g)( t) (h) and (h)( t) (g), thus one can get

T T fñ(−t) = |T F˜ T| = |F˜| = fñ, gñ(−t) = |T H˜ T| = |F˜| = h˜ n. (94)

gñ(−t) h˜ n From the transformation (32), it immediately reaches un(−t) = = = vn. Furthermore, through the fñ(−t) fñ variable transformations [ ( ) ] ψ 1 a2 − 1 1 u = √n exp 2i − 1 t , α = , c = , (95) n α a2 γa2 a2 with the complex constant γ, we obtain the reverse time discrete symmetric IDNLS equation (6). Finally, we get the following theorem about the solution of the nonlocal IDNLS equation (6). Theorem 3.3 The nonlocal IDNLS equation (6) has the solution √ 2 − ( ) a 1 −2i 1 −1 t gñ ψ = e a2 , (96) n γ 2 a fñ

11 where tau functions are given by

AB A B f˜ = , g˜ = (g) , (97) n n CD CD(g) 2N×2N 2N×2N with the block matrices    [ ]   n η′   n η′   q e i   − q e i  A =  1 + δ 1 i  , A =  1 + δ 1 qi(1 api) i  ,  i j ξ′  (g)  i j ξ′  pi − q j pi − q j pne j pi − q j pi − q j pi(1 − aqi) pne j j N×N j N×N  η′ −   [ ] η′ −   n j( t)   n j( t)   1 1 q e   1 1 a − q q e  D =  + δ j  , D =  + δ i j  ,  i j n ξ′(−t)  (g)  i j n ξ′(−t)  p j − qi p j − qi p e i p j − qi p j − qi a − pi p e i i N×N i N×N and ( ) ( ) B = 1 , C = 1 . − − pi p j 1 N×N 1 qiq j N×N ( ) ( ) ′ 1 t ′ ′ 1 t ′ ′ ′ Here ξ = i pi + + ξ and η = i qi + + η , pi, qi ξ ,η (i = 1, 2 · ·· , N) and a are arbitrary complex i pi a i,0 i qi a i,0 i,0 i,0 constants, these parameters need to satisfy the constraint conditions

apiqi + a − pi − qi = 0. (98)

For example, by taking N = 1, tau functions for the nonlocal IDNLS equation (6) are written as:   n η′ n η′ (−t) 2n η′ +η′ (−t)  q e 1 q e 1 q e 1 1   1 1 1  f = ∆ + ∆ 1 + ′ + ′ + ′ ′  , (99) n 1 2 n ξ n ξ (−t) 2n ξ +ξ (−t) p e 1 p e 1 p e 1 1  1 1 1  n η′ n η′ (−t) 2n η′ +η′ (−t)  q e 1 q e 1 q e 1 1   1 1 1 1  g = ∆ + ∆ 1 + P ′ + ′ + ′ ′  , (100) n 1 2 1 n ξ n ξ (−t) 2n ξ +ξ (−t) 1 P 1 1 1 p1e 1 p1e p1 e where 1 1 a − p ∆ = , ∆ = , P = 1 , 1 2 − 2 − 2 p − q 2 1 a − q (p1 1)(q1 1) ( 1 1) 1 ( ) ( ) ′ 1 t ′ ′ 1 t ′ ′ ′ and ξ = i p1 + + ξ and η = i q1 + + η , p1, q1 ξ ,η and a are arbitrary complex constants, they 1 p1 a 1,0 1 q1 a 1,0 1,0 1,0 need to satisfy the constraint condition ap1q1 + a − p1 − q1 = 0.

4. Summary and discussions

In this paper, we study Gram determinant solutions for local and nonlocal IDNLS equations via the pair reduction. Starting from the coupled system of the TL’s BT and D2DTL equation, the generalized IDNLS equation with the single Casorati determinant solution is derived through the dimensional reduction. According to the gauge invariance of the bilinear form for the generalized IDNLS equation, the single Casorati determinant solution is changed as two kinds of Gram determinant solutions. Based on the different forms of the Gram determinant solution, the pair constraint conditions for wave numbers are imposed and then solutions of local and nonlocal IDNLS equations are constructed. These solutions corresponds to the even number of soliton respectively.

Acknowledgment

J.C. acknowledges support from National Natural Science Foundation of China (No.11705077). B.F.F. was partially supported by NSF under Grant No. DMS-1715991, NSF of China under Grant No.11728103 and the COS Research Enhancement Seed Grants Program at The University of Texas Rio Grande Valley. Y.J. is supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No.LY14A010016.

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