Soliton solutions and their dynamics in reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schrödinger equations
SSoliton solutions and their dynamics in reverse-spaceand reverse-space-time nonlocal discrete derivativenonlinear Schr¨odinger equations
Gegenhasi ∗ , Yue-Chen JiaSchool of Mathematical Science, Inner Mongolia University,No.235 West College Road, Hohhot, Inner Mongolia 010021, PR CHINAJune 9, 2020 Abstract
In this paper, we introduce the reverse-space and reverse-space-time nonlocal discrete derivative non-linear Schr¨odinger (DNLS) equations through the nonlocal symmetry reductions of the semi-discreteGerdjikov-Ivanov equation. The muti-soliton solutions of two types of nonlocal discrete derivative non-linear Schr¨odinger equations are derived by means of the Hirota bilinear method and reduction approach.We also investigate the dynamics of soliton solutions and reveal the rich soliton structures in the reverse-space and reverse-space-time nonlocal discrete DNLS equations. Our investigation shows that the solitonsof these nonlocal equations often breathe and periodically collapse for some soliton parameters, but re-main nonsingular for other range of parameters.
KEYWORDS:
Nonlocal discrete derivative nonlinear Schr¨odinger equations, Hirota bilinear method, Soli-ton solution, Soliton dynamics
MSC:
Since Ablowitz and Musslimani proposed continuous and discrete reverse-space, reverse-time and reverse-space-time nonlocal nonlinear integrable equations by introducing new nonlocal symmetry reductions of theAKNS scattering problem and Ablowitz-Ladik scattering problem [1, 2, 3], the nonlocal integrable equationshave triggered renewed interest in integrable systems. A variety of mathematical methods such as inversescattering methods [1, 2, 3, 4, 5], Darboux transformation methods [6, 7, 8], Hirota’s bilinear method andKP hierarchy reduction method [9, 10, 11, 12, 13, 14] have been applied to study the nonlocal integrableequations. The nonlocal integrable equations possess some specific solution behaviors, such as finite-timesolution blowup[1, 15], the simultaneous existence of solitons and kinks[16], the simultaneous existence ofbright and dark solitons[1, 4], and distinctive multisoliton patterns[17].In [18], the author proposed an integrable semi-discrete Gerdjikov-Ivanov equation (cid:40) iq n,t + ( q n +1 + q n − − q n ) − q n ( q n +1 + q n − ) ( r n +1 − r n + q n r n r n +1 ) = 0 ,ir n,t − ( r n +1 + r n − − r n ) + r n ( r n +1 + r n − ) ( q n − − q n + r n q n q n − ) = 0 , (1.1)where q n = q ( n, t ) , r n = r ( n, t ) are complex functions on Z × R . The Miura map u n = q n , v n = r n +1 − r n + q n r n r n +1 and another Miura map u n = q n − − q n + r n q n q n − , v n = r n connect the semi-discrete Gerdjikov-Ivanov equation (1.1) with the coupled discrete nonlinear Schr¨odinger equation proposed by Ablowitz and ∗ Corresponding author. E-mail: [email protected] a r X i v : . [ n li n . S I] J un adik (cid:40) iu n,t + ( u n +1 + u n − − u n ) − u n v n ( u n +1 + u n − ) = 0 ,iv n,t − ( v n +1 + v n − − v n ) + u n v n ( v n +1 + v n − ) = 0 . (1.2)The semi-discrete Gerdjikov-Ivanov equation (1.1) has been solved by the inverse scattering method[18].However, the Hirota bilinear formalism of Eq.(1.1) has not been reported yet. In this paper, we presentthe bilinear form of the semi-discrete Gerdjikov-Ivanov equation (1.1) and obtain its one-, two- and three-soliton solutions via Hirota bilinear method. It is known that the semi-discrete Gerdjikov-Ivanov equation(1.1) admits the local reduction of complex conjugation r n = ± iq ∗ n − . In this paper, we introduce twonew nonlocal symmetry reductions r n = σq ∗− n , σ = ± r n = σq − n ( − t ) , σ = ± iq n,t + ( q n − + q n +1 − q n ) + σq n ( q n − + q n +1 )( q ∗− n − q ∗− n − − σq ∗− n q n q ∗− n − ) = 0 , (1.3)and iq n,t + ( q n − + q n +1 − q n ) + σq n ( q n − + q n +1 )( q − n ( − t ) − q − n − ( − t ) − σq − n ( − t ) q n q − n − ( − t )) = 0 , (1.4)respectively. We derive one-, two- and three-soliton solutions for reverse-space discrete DNLS equation (1.3)and reverse-space-time discrete DNLS equation (1.4), and study the dynamics of these soliton solutions.The paper is organized as follows. In Section 2, we derive one-, two- and three-soliton solutions for thesemi-discrete Gerdjikov-Ivanov equation (1.1) by applying the Hirota bilinear method. In Section 3, one-,two- and three-soliton solutions for the reverse-space discrete DNLS equation (1.3) are derived through thereduction approach and dynamics of these solitons are discussed. In Section 4, we derive one-, two- andthree-soliton solutions for the reverse-space-time discrete DNLS equation (1.4) via the reduction approachand investigate rich dynamics of soliton solutions. We end this paper with a conclusion and discussion inSection 5. In this section, we first bilinearise the semi-discrete Gerdjikov-Ivanov equation (1.1) and derive its one-,two- and three-soliton solutions via the Hirota bilinear method[19].Through the dependent variable transformations q n = g n f n , r n = − h n s n , (2.1)Eq.(1.1) is transformed into the bilinear form iD t f n • g n = f n − g n +1 + f n +1 g n − − f n g n ,iD t h n • s n = h n +1 s n − + h n − s n +1 − h n s n ,g n h n − f n s n + f n − s n +1 = 0 ,g n h n +1 + f n s n +1 − f n +1 s n = 0 , (2.2)where the bilinear operator D mx D nt is defined by [19] D mx D nt f • g = ∂ m ∂y m ∂ n ∂s n f ( x + y, t + s ) g ( x − y, t − s ) | s =0 ,y =0 . According to Hirota bilinear method, in order to construct one-soliton solution, we expand the functions g n , f n , h n and s n with a small parameter ε as g n = εg (1) n , h n = εh (1) n , f n = 1 + ε f (2) n , s n = 1 + ε s (2) n . (2.3)2y inserting expansions (2.3) into bilinear equations (2.2), we obtain the coefficient of ε − ig (1) n,t = g (1) n +1 + g (1) n − − g (1) n , ih (1) n,t = h (1) n +1 + h (1) n − − h (1) n . (2.4)If we take the solution of linear differential-difference equations (2.4) in the form g (1) n = e ξ , h (1) n = e η , (2.5)with ξ = kn + ωt + δ, η = ln + ρt + α , then we yield the dispersion relations ω = 4 i sinh k , ρ = − i sinh l . (2.6)The coefficient of ε gives g (1) n h (1) n − s (2) n − f (2) n + s (2) n +1 + f (2) n − = 0 , g (1) n h (1) n +1 + s (2) n +1 + f (2) n − s (2) n − f (2) n +1 = 0 . (2.7)We obtain a solution of linear differential-difference equations (2.7) in the exponential form f ,n = Ae ξ + η , s ,n = Be ξ + η , (2.8)where A = e l −
14 sinh k + l , B = e − k −
14 sinh k + l . (2.9)It can be verified that the coefficients of ε , ε are automatically satisfied if we substitute (2.5) and (2.8) intothem. Therefore, one-soliton solution of the semi-discrete Gerdjikov-Ivanov equation (1.1) is given by q n = e ξ Ae ξ + η , r n = − e η Be ξ + η , (2.10)with ξ = kn + (4 i sinh k ) t + δ, η = ln − (4 i sinh l ) t + α, A = e l −
14 sinh k + l and B = e − k −
14 sinh k + l . Here k, l, δ and α are arbitrary complex parameters.For two-soliton solution, we take g n = εg (1) n + ε g (3) n , h n = εh (1) n + ε h (3) n , f n = 1 + ε f (2) n + ε f (4) n , s n = 1 + ε s (2) n + ε s (4) n . (2.11)When we insert expansions (2.11) into (2.2) and consider the coefficients of ε , we derive g (1) n = e ξ + e ξ , h (1) n = e η + e η , with ξ j = k j n + ω j t + δ j , η j = l j n + ρ j t + α j for j = 1 ,
2, and the dispersion relations ω j = 4 i sinh k j , ρ j = − i sinh l j , j = 1 , . (2.12)From the coefficient of ε , we derive f (2) n = e ξ + η + α , + e ξ + η + α , + e ξ + η + α , + e ξ + η + α , ,s (2) n = e ξ + η + δ , + e ξ + η + δ , + e ξ + η + δ , + e ξ + η + δ , , where e α m,j = e l j −
14 sinh k m + l j , e δ m,j = e − k m −
14 sinh k m + l j , m, j = 1 , . (2.13)The coefficient of ε gives g (3) n = ˆ A e ξ + ξ + η + ˆ A e ξ + ξ + η , h (3) n = ˆ B e ξ + η + η + ˆ B , e ξ + η + η , A m = (cid:0) e l m − (cid:1) sinh k − k k + l m sinh k + l m , ˆ B m = (cid:0) e − k m − (cid:1) sinh l − l k m + l sinh k m + l , m = 1 , . (2.14)From the coefficient of ε , we derive f (4) n = M e ξ + ξ + η + η , s (4) n = N e ξ + ξ + η + η , where M = (cid:0) e l − (cid:1) (cid:0) e l − (cid:1) sinh k − k sinh l − l
16 sinh k + l sinh k + l sinh k + l sinh k + l , N = (cid:0) e − k − (cid:1) (cid:0) e − k − (cid:1) sinh k − k sinh l − l
16 sinh k + l sinh k + l sinh k + l sinh k + l . (2.15)It can be verified the coefficients of ε , ε , ε , ε are automatically satisfied. Therefore, two-soliton solutionof the semi-discrete Gerdjikov-Ivanov equation (1.1) is given by q n = e ξ + e ξ + ˆ A e ξ + ξ + η + ˆ A e ξ + ξ + η e ξ + η + α , + e ξ + η + α , + e ξ + η + α , + e ξ + η + α , + M e ξ + ξ + η + η , (2.16) r n = − e η + e η + ˆ B e ξ + η + η + ˆ B e ξ + η + η e ξ + η + δ , + e ξ + η + δ , + e ξ + η + δ , + e ξ + η + δ , + N e ξ + ξ + η + η , (2.17)with ξ m = k m n + (4 i sinh k m ) t + δ m , η m = l m n − (4 i sinh l m ) t + α m ( m = 1 , α m,j , δ m,j , A m , B m , M, N are given by (2.13)-(2.15). Here k m , l m , δ m and α m ( m = 1 ,
2) are arbitrary complexparameters.For three-soliton solution, we take g n = εg (1) n + ε g (3) n + ε g (5) n , h n = εh (1) n + ε h (3) n + ε h (5) n ,f n = 1 + ε f (2) n + ε f (4) n + ε f (6) n , s n = 1 + ε s (2) n + ε s (4) n + ε s (6) n . (2.18)By substituting expansions (2.18) into bilinear equations (2.2) and considering the coefficients of ε , we derive g (1) n = e ξ + e ξ + e ξ , h (1) n = e η + e η + e η , with ξ j = k j n + ω j t + δ j , η j = l j n + ρ j t + α j for j = 1 , ,
3, and the dispersion relations ω j = 4 i sinh k j , ρ j = − i sinh l j , j = 1 , , . (2.19)The coefficient of ε gives f (2) n = (cid:88) ≤ m,j ≤ e ξ m + η j + α m,j , s (2) n = (cid:88) ≤ m,j ≤ e ξ m + η j + δ m,j , where e α m,j = e l j −
14 sinh k m + l j , e δ m,j = e − k m −
14 sinh k m + l j , m, j = 1 , , . (2.20)The coefficient of ε gives g (3) n = (cid:88) ≤ m 16 sinh k m + l µ sinh k m + l ν sinh k j + l µ sinh k j + l ν , (2.22) N m,j,µ,ν = (cid:0) e − k m − (cid:1) (cid:0) e − k j − (cid:1) sinh k m − k j sinh l µ − l ν 16 sinh k m + l µ sinh k m + l ν sinh k j + l µ sinh k j + l ν . (2.23)The coefficient of ε gives g (5) n = (cid:88) (cid:54) m 16 sinh k + l m sinh k + l j sinh k + l m sinh k + l j sinh k + l m sinh k + l j , (2.24)˜ B m,j = (cid:0) e − k m − (cid:1) (cid:0) e − k j − (cid:1) sinh k m − k j sinh l − l sinh l − l sinh l − l 16 sinh k m + l sinh k j + l sinh k m + l sinh k j + l sinh k m + l sinh k j + l . (2.25)The coefficient of ε gives f (6) n = Je ξ + ξ + ξ + η + η + η , s (6) n = Ke ξ + ξ + ξ + η + η + η , where J = (cid:81) p ∈{ , , } ( e l p − (cid:81) m,j ∈{ , , } m 3) and the coefficients α m,j , δ m,j , ˜ A m,j , ˜ B m,j , A m,j,µ , B s,i,j , M i,j,s,t , N i,j,s,t , J, K are given by (2.20-2.27). Here k i , l i , δ i and α i ( i = 1 , , 3) are arbi-trary complex parameters. In this section, we derive one-, two-, three-soliton solutions for the reverse-space DNLS equation (1.3) byfinding the constraint conditions on the parameters of one-, two-, three-soliton solutions of the semi-discreteGerdjikov-Ivanov equation (1.1) to satisfy the the reduction formula r n = σq ∗− n .5 .1 One-soliton solutions From one-soliton solution (2.10) and reduction formula r n = σq ∗− n , we have − e ln + ρt + α Be ( k + l ) n +( ω + ρ ) t + δ + α = σe − k ∗ n + ω ∗ z + δ ∗ A ∗ e − ( k ∗ + l ∗ ) n +( ω ∗ + ρ ∗ ) t + δ ∗ + α ∗ , (3.1)which yields the constraint conditions on four free paramaters k, l, δ, α :(1) l = − k ∗ , (2) ρ = ω ∗ , (3) e α = − σe δ ∗ , (4) B = A ∗ , (5) k + l = − ( k ∗ + l ∗ ) , (6) ρ + ω = ω ∗ + ρ ∗ , (7) e δ + α = e δ ∗ + α ∗ . (3.2)Utilizing the dispersion relation (2.6) and (2.9), Eq.(3.2) can be reduced to the following two constraints(1) l = − k ∗ , (2) e α = − σe δ ∗ . (3.3)Therefore, the reverse-space discrete DNLS equation (1.3) has the following form of one soliton solution q n = e kn +(4 i sinh k ) t + δ − Aσe ( k − k ∗ ) n +4 i (sinh k − sinh k ∗ ) t +( δ + δ ∗ ) , (3.4)where A = e − k ∗ − 14 sinh k − k ∗ and k, δ are arbitrary complex parameters.By letting k = a + bi, δ = c + di, A = L + M i , we obtain | q n | = e an e − R + e R ( L + M ) − σ √ L + M cos(2 bn + γ ) , (3.5)where R = c − b ) sinh( a ) t and γ is determined by sin( γ ) = M √ L + M , cos( γ ) = L √ L + M . In the spaecialcase a = 0, (3.5) becomes | q n | = 1 e − c + e c ( L + M ) − σ √ L + M cos(2 bn + γ ) , (3.6)which is a spatial periodical solution with the period πb . By taking parameters as k = 2 i, δ = 3 + 4 i, σ = − , the spatial periodical solution (3.6) is illustrated in ( a ) of Fig.1.If a (cid:54) = 0 , then one-soliton solution (3.4) would breathe and periodically collapse in n at time t = c + ln( L M b ) sinh( a ) and its amplitude | q n | changes as | q n | = √ L + M e an − σ cos(2 bn + γ )) . (3.7)When b (cid:54) = 0 , this soliton periodically collapses in n with period πb and its amplitude grows or decaysexponentially (depending on the sign of a ), which are shown in ( a ) and ( b ) of Fig.2 by choosing the parametersas k = − . − . i, δ = 1 + πi, σ = − , and k = 0 . . i, δ = 1 + πi, σ = − , respectively.We obtain another type of one-soliton solution for the reverse-space discrete DNLS equation (1.3) by thecross multiplication reduction. Applying the cross multiplication on Eq.(3.1), we obtain − e ln + ρt + α (1 + A ∗ e − ( k ∗ + l ∗ ) n +( ω ∗ + ρ ∗ )+ δ ∗ + α ∗ ) = σe − k ∗ n + ω ∗ t + δ (1 + Be ( k + l ) n +( ω + ρ ) t + δ + α ) , (3.8)from which we derive the conditions (1) k = k ∗ , l = l ∗ (2) e δ + δ ∗ = − σB , e α + α ∗ = − σA ∗ , (3.9)6n which A = e l − 14 sinh k + l and B = e − k − 14 sinh k + l . Setting δ = a + bi , α = c + di , then according to the Eq.(3.9),we obtain (1) e a = (cid:114) − σB , (2) e c = (cid:114) − σA , (3.10)where a, b, c, d, k, l are real.Therefore, another type of one soliton solution for Eq.(1.3) is given by q ( n, t ) = e kn +4 i sinh k t + bi √− σB (1 + (cid:113) AB e ( k + l ) n +4 i (sinh k t − sinh l ) t +( b + d ) i ) , (3.11)where b, d, k, l are free real parameters. The corresponding | q n | is | q n | = e kn +2 a A e k + l ) n +2( a + c ) + 2 A cos( R ) e ( k + l ) n +( a + c ) , (3.12)where R = 4(sinh k − sinh l ) t + ( b + d ). From (3.12), we derive one-soliton solution (3.11) breathes andperiodically collapses in time at position n = ln AB k + l ) , in which the condition ln AB k + l ) ∈ Z should be satisfied.The period of this collapse is π k − sinh l ) .The graph of one soliton solution (3.11) is depicted in ( b ) of Fig.1 by taking the parameters: σ = − , k = ln(1 − e − . ) , l = 0 . , b = 1 , d = 1 . (a) (b) Fig. 1: One-soliton solution for Eq.(1.3): (a) Nonsingular spatial periodic solution, (b) solution breathingand periodically collapsing in time. From the two-soliton solution (2.16-2.17) and reduction formula r n = σq ∗− n , we have − e η + e η + ˆ B e ξ + η + η + ˆ B e ξ + η + η e ξ + η + δ , + e ξ + η + δ , + e ξ + η + δ , + e ξ + η + δ , + N e ξ + ξ + η + η = σ e ¯ ξ ∗ + e ¯ ξ ∗ + ˆ A ∗ e ¯ ξ ∗ +¯ ξ ∗ +¯ η ∗ + ˆ A ∗ e ¯ ξ ∗ +¯ ξ ∗ +¯ η ∗ e ¯ ξ ∗ + ¯ η ∗ + α ∗ , + e ¯ ξ ∗ +¯ η ∗ + α ∗ , + e ¯ ξ ∗ +¯ η ∗ + α ∗ , + e ¯ ξ ∗ +¯ η ∗ + α ∗ , + M e ¯ ξ ∗ +¯ ξ ∗ +¯ η ∗ +¯ η ∗ , (3.13)7 a) (b) Fig. 2: One-soliton solution periodically collapsing in space: (a)Solution with exponentially growing ampli-tude, (b)Solution with exponentially decaying amplitude.where ¯ ξ j = − k j n + ω j t + δ j , ¯ η j = − l j n + ρ j t + α j ( j = 1 , . Eq.(3.13) yields the constraint conditions on theeight paramaters k j , l j , δ j , α j ( j = 1 , l j = − k ∗ j , j = 1 , , (2) a j = ω ∗ j , j = 1 , , (3) e α j = − σe δ ∗ j , j = 1 , , (4) ˆ B j = ˆ A ∗ j , j = 1 , , (5) k + l + l = − ( k ∗ + k ∗ + l ∗ ) , k + l + l = − ( k ∗ + k ∗ + l ∗ ) , (6) e α ∗ m,j = e δ j,m , m, j = 1 , , (7) ω + ρ + ρ = − ( ω ∗ + ω ∗ + ρ ∗ , ) , ω + ρ + ρ = − ( ω ∗ + ω ∗ + ρ ∗ ) , (8) N = M ∗ . (3.14)Utilizing the dispersion relations (2.12) and Eqs.(2.13-2.15), Eq.(3.14) can be reduced to the following fourconditions (1) l j = − k ∗ j , (2) e α j = − σe δ ∗ j , j = 1 , . (3.15)Therefore, the two-soliton solution for the reverse-space discrete DNLS equation (1.3) is given by (2.16) withconstraints of parameters (3.15). The graph of this two-soliton solution is depicted in Fig.3 and Fig.4 bytaking the parameters as k = 0 . i, k = 0 . i, δ = 1 + 2 i, δ = i, σ = − , and ( a ) k = 0 . . i, k = − . − . i, δ = 0 , δ = 0 , σ = 1 , ( b ) k = 0 . . i, k = − . − . i, δ = 0 , δ = 0 , σ = 1 , respectively.We derive another type of two-soliton solution for the reverse-space discrete DNLS equation (1.3) viathe cross multiplication reduction. Applying the cross multiplication on (3.13). we obtain the followingconstraints on eight paramaters k j , l j , δ j , α j ( j = 1 , k j = k ∗ j , l j = l ∗ j ( j = 1 , , (2) e δ + δ ∗ = − ˆ B σN , (3) e δ + δ ∗ = − ˆ B σN , (4) e α + α ∗ = − σ ˆ A ∗ M ∗ , (5) e α + α ∗ = − σ ˆ A ∗ M ∗ . (3.16)We suppose δ j = a j + b j i , α j = x j + y j i ( j = 1 , a j , b j , x j , y j ( j = 1 , 2) are real. According to (3.16),we obtain (1) e a = 2 (cid:115) sinh k + l sinh k + l σ (1 − e − k ) sinh k − k , (2) e a = 2 (cid:115) sinh k + l sinh k + l σ (1 − e − k ) sinh k − k , (3) e x = 2 (cid:115) sinh k + l sinh k + l σ (1 − e l ) sinh l − l , (4) e x = 2 (cid:115) sinh k + l sinh k + l σ (1 − e l ) sinh l − l . (3.17)8 a) (b) Fig. 3: Two-soliton solution for Eq.(1.3): (a)Nonsingular periodic two-soliton, (b)The density profiles of (a). (a) (b) Fig. 4: Two-soliton solution for Eq.(1.3): (a)Two nonsingular solitons with changing amplitude moving inopposite directions, (b)Elastic collision of two soliton.Therefore, another type of two-soliton solution for the reverse-space discrete DNLS equation (1.3) is givenby (2.16) with constraints of parameters (3.17). We illustrate this two-soliton in Fig.5 by taking k = 0 . , k = 0 . , l = 0 . , l = 0 . , b = 0 , b = 0 , y = 0 , y = 0 , σ = 1 . Similar to one- and two- soliton solution for the reverse-space discrete DNLS equation (1.3), we obtain thefollowing conditions on the parameters of three-soliton solution (2.28-2.29) to satisfy the nonlocal reduction r n = σq ∗− n : l j = − k ∗ j , ρ j = ω ∗ j , e α j = − σe δ j , j = 1 , , e δ m,j = e α ∗ j,m , m, j = 1 , , K = J ∗ ;˜ B m,j = ˜ A ∗ m,j , B µ,m,j = A ∗ m,j,µ , m, j, µ = 1 , , , m < j ; N m,j,µ,ν = M ∗ µ,ν,m,j , m, j, µ, ν = 1 , , , m < j, µ < ν. (3.18)Utilizing the dispersion relations (2.19) and Eqs.(2.20-2.27), Eq.(3.18) can be reduced to the following sixconditions l j = − k ∗ j , e α ∗ j = − σe δ j , j = 1 , , . (3.19)9 a) (b) Fig. 5: Two-soliton solution for Eq.(1.3): (a) periodically breathing bounded two-soliton, (b)The densityprofiles of (a).Therefore, the 3-soliton solution of the nonlocal discrete DNLS (1.3) is given by (2.28) with constraints ofparameters (3.19). we choose parameters in three-soliton solution as k = 0 . i, k = 0 . i, k = 0 . i, δ = i, δ = i, δ = i, σ = − , and the corresponding three-soliton is shown in Fig.6 . (a) (b) Fig. 6: Three-soliton solution for Eq.(1.3): (a)bounded periodic three-soliton, (b)The density profiles of (a). In this section, we derive one-, two-, three-soliton solutions of the reverse-space-time discrete DNLSequation (1.4) by finding the constraint conditions on the parameters of one-, two-, three-soliton solutionsof the semi-discrete Gerdjikov-Ivanov equation (1.1) to satisfy the the reduction formula r n = σq − n ( − t ). From one-soliton solution (2.10) and reduction formula r n = σq − n ( − t ), we have − e ln + ρt + α Be ( k + l ) n +( ω + ρ ) t + δ + α = σe − kn − ωt + δ Ae − ( k + l ) n − ( ω + ρ ) t + α + δ . (4.1)10y applying the cross multiplication on (4.1), we obtain − (cid:0) e ln + ρt + α + Ae − kn − ωt + δ +2 α (cid:1) = σe − kn − ωt + δ + Bσe ln + ρt +2 δ + α , (4.2)form which we derive Ae α = − σ, Be δ = − σ, (4.3)which yields e α = (cid:113) − σA and e δ = (cid:113) − σB . Therefore, one soliton solution for the reverse-space-time discreteDNLS equation (1.4) is given by q n = e kn +(4 i sinh k ) t √− σB (1 + (cid:113) AB e ( k + l ) n +4 i (sinh k − sinh l ) t ) , (4.4)where k, l are free complex parameters. By setting k = a + bi, c + di, (cid:113) AB = R + Ii , the corresponding | q n | is given by | q n | = 1 | B | ( e − ζ + ( R + I ) e ζ + 2 √ R + I cos( L + γ ) e ζ − ζ ) , (4.5)where ζ = an − a ) sin( b ) t, ζ = cn +2 sinh( c ) sin( d ) t , L = ( b + d ) n +2(cosh( a ) cos( b ) − cosh( c ) cos( d )) t, cos( γ ) = R √ R + I , sin( γ ) = L √ R + I . Case I. b = d = 0 . In this case, | q n | can be written as | q n | = 1 | B | ( e − an + R e cn + 2 | R | cos(2(cosh( a ) − cosh( c )) t + γ ) e ( c − a ) n ) , (4.6)from which we derive that this soliton breathes and periodically collapses in t with period π cosh( a ) − cosh( c ) at position n = − ln( ec − e − a − )2( a + c ) where the conditions ac < − ln | ec − e − a − | a + c ∈ Z should be satisfied. At n = − ln( ec − e − a − )2( a + c ) , the amplitude of the soliton changes as | q n | = 1 | B | ( | R | aa + c + | R | − aa + c + 2 | R | aa + c cos(2(cosh( a ) − cosh( c )) t + γ )) . (4.7)By taking k = ln 23 , l = ln 3 , σ = − , this soliton is illustrated in (a) of Fig.7.Case II. a = c = 0 . In this case, the | q n | becomes | q n | = 1 | B | (1 + R + I + 2 √ R + I cos((( b + d ) n + 2(cos( b ) − cos( d )) t + γ ) . (4.8)When R + I (cid:54) = 1 , this soliton is bounded and periodic which is shown in (b) of Fig.7 by taking k = i, l = 0 . i, σ = 1 . Case III. a, c are not simultaneously zero and b, d are not simultaneously zero.In this case, this soliton moves at velocity V = a ) sin( b ) − sinh( c ) sin( d )) a + c on the line n = V t − a + c ) ln( R + I ) where the amplitude | q n | changes as | q n | = ( R + I ) − aa + c | B | e (cid:37)t t + ϑ ) , (cid:37) = aV − a ) sin( b ) , Ω = ( b + d ) V + 2(cosh( a ) cos( b ) − cosh( c ) cos( d )) , ϑ = γ − b + d a + c ) ln( R + I ) . When Ω (cid:54) = 0 , this soliton periodically collapses with period π Ω , and when (cid:37) (cid:54) = 0 , the amplitude of the solitongrows or decays exponentially (depending on the sign of (cid:37) ) which are illustrated in (a) and (b) of Fig.8 bytaking parameters as k = 0 . − i, l = 0 . − . i, σ = 1 , and k = 0 . i, l = 0 . . i, σ = 1 , respectively. (a) (b) Fig. 7: One-soliton solution for the reverse-space-time discrete DNLS equation (1.4): (a)One-soliton breath-ing and periodically collapsing in time, (b) bounded periodic one-soliton. (a) (b) Fig. 8: Periodically collapsing one-soliton solution for Eq.(1.4): (a) Solution with exponentially growingamplitude, (b) Solution with exponentially decaying amplitude. From the two-soliton solution (2.16-2.17) and reduction formula r n = σq − n ( − t ), we have − e η + e η + ˆ B e ξ + η + η + ˆ B e ξ + η + η e ξ + η + δ , + e ξ + η + δ , + e ξ + η + δ + e ξ + η + δ , + N e ξ + ξ + η + η = σ e ξ − + e ξ − + ˆ A e ξ − + ξ − + η − + ˆ A e ξ − + ξ − + η − e ξ − + η − + α , + e ξ − + η − + α , + e ξ − + η − + α , + e ξ − + η − + α , + M e ξ − + ξ − + η − + η − , (4.9)12here ξ − j = − k j n − ω j t + δ j , η − j = − l i n − ρ j t + α j ( j = 1 , . Applying the cross multiplication, we getˆ B e α j + α ,j +2 δ + ˆ B e α j + α ,j +2 δ + σN ˆ A j e α j +2 δ +2 δ + σe δ + δ ,j + σe δ + δ ,j + 1 = 0 , j = 1 , , ˆ A e δ j + δ j, +2 α + ˆ A e δ j + δ j, +2 α + σM ˆ B j e δ j +2 α +2 α + σe α + α j, + σe α + α j, + 1 = 0 , j = 1 , ,σ ˆ A λ e δ ν + δ ν,µ + e α β,λ = 0 , σ ˆ B λ e α ν + α µ,ν + e δ λ,β = 0 , λ, ν ∈ { , } ; µ ∈ { , }\{ λ } ; β ∈ { , }\{ ν } ,σ ˆ A m + M e α j = 0 , σ ˆ B m + N e δ j = 0 , ≤ j (cid:54) = m ≤ . (4.10)Utilizing the dispersion relations (2.12) and Eqs.(2.13-2.15), Eq.(4.10) can be reduced to the following fourconditions M e α j = − σ ˆ A m , N e δ j = − σ ˆ B m , ≤ j (cid:54) = m ≤ , (4.11)from which we have e α j = 2 (cid:118)(cid:117)(cid:117)(cid:116) sinh k + l j sinh k + l j σ (1 − e l j ) sinh l − l , e δ j = 2 (cid:118)(cid:117)(cid:117)(cid:116) sinh k j + l sinh k j + l σ (1 − e − k j ) sinh k − k , j = 1 , , (4.12)where k j , l j ( j = 1 , 2) are arbitrary complex parameters. Therefore, (2.16) with constraints of parameters(4.12) gives two-soliton solution for the reverse-space-time discrete DNLS equation (1.4). A periodicallybreathing but not collapsing two-soliton solution which is asymmetric in n is depicted in Fig.9 by taking theparameters as k = 0 . , k = 0 . , l = 0 . , l = 0 . , σ = 1 . The collisions of two bounded soliton are displayed in ( a ) and ( b ) of Fig.10 by choosing parameters as (a) (b) Fig. 9: Two-soliton solution for the reverse-space-time discrete DNLS equation (1.4): (a)Breathing 2-soliton,(b)The density profiles of (a). k = 0 . . i, k = 0 . − . i, l = 0 . − . i, l = 0 . . i, σ = 1 , and k = 0 . . i, k = 0 . − . i, l = 0 . − . i, l = 0 . . i, σ = 1 , respectively. By applying cross multiplication on the three-soliton solution (2.28-2.29) with the nonlocal reduction r n ( t ) = σq − n ( − t ), we obtain 126 constraints on parameters which are given in Appendix A.13 a) (b) Fig. 10: Two-soliton solution for the reverse-space-time discrete DNLS equation (1.4): (a)Collision of twobounded soliton with exponentially decaying amplitudes, (b)Elastic collision of two soliton.Applying the dispersion relations (2.19) and Eqs.(2.20-2.27), Eqs.(A.1-A.6) can be reduced to the follow-ing six constraints: σ ˜ A m,p = − Je α j , σ ˜ B m,p = − Ke δ j , j ∈ { , , } , m, p ∈ { , , }\{ j } , p > m, which yields e α j = 2 (cid:118)(cid:117)(cid:117)(cid:116) sinh k + l j sinh k + l j sinh k + l j σ (1 − e l j ) sinh lj − lm sinh lj − lp , j ∈ { , , } , m, p ∈ { , , }\{ j } , p > m, (4.13) e δ j = 2 (cid:118)(cid:117)(cid:117)(cid:116) sinh k j + l sinh k j + l sinh k j + l σ (1 − e − k j ) sinh kj − km sinh kj − kp , j ∈ { , , } , m, p ∈ { , , }\{ j } , p > m, (4.14)where k j , l j ( j = 1 , , 3) are arbitrary complex parameters. Therefore, Eq.(2.28) with constraints on param-eters (4.13-4.14) gives three-soliton solution for the reverse-space-time discrete DNLS equation (1.4). Thebounded three-soliton solution which breathes periodically in t is displayed in Fig.11 by taking parametersin this three-soliton solution as k = 0 . , k = 0 . , k = 0 . , l = 0 . , l = 0 . , l = 0 . , σ = − . The interactions of three bounded solitons are displayed in Fig.12 by takeing the parameters as k = 0 . 15 + 0 . i, k = 0 . 24 + 0 . i, k = 0 . − . i, l = 0 . − . i, l = 0 . − . i, l = 0 . 24 + 0 . i, σ = − . In this paper, we proposed the reverse-space and reverse-space-time nonlocal discrete DNLS equations(1.3) and (1.4), and derived their one-, two- and three-soliton solutions via Hirota bilinear method andreduction approach. The dynamics of soliton solutions are discussed and rich soliton structures in the reverse-space and reverse-space-time nonlocal discrete DNLS equations are revealed. Our investigation shows thatthe solitons of these nonlocal equations often breathe and periodically collapse for some soliton parameters,but remain bounded for other range of parameters.Now we investigate the continuous limit for the reverse-space nonlocal discrete DNLS equation (1.3), thereverse-space-time nonlocal discrete DNLS equation (1.4) and their one-soliton solutions. If we take q n = εQ ( x, τ ) , x = nε , τ = ε t, a) (b) Fig. 11: Three-soliton solution for the reverse-space-time discrete DNLS equation (1.4): (a)Periodicallybreathing bounded three-soliton solution, (b)The density profiles of (a). (a) (b) Fig. 12: Three-soliton solution for the reverse-space-time discrete DNLS equation (1.4): (a)Collision ofbounded three soliton, (b)The density profiles of (a).then as ε → 0, Eq. (1.3) and Eq. (1.4) converge to the reverse-space and reverse-space-time nonlocal DNLSequations iQ τ + Q xx − σQ Q ∗ x ( − x ) − Q Q ∗ ( − x ) = 0 , (5.1)and iQ τ + Q xx − σQ Q x ( − x, − τ ) − Q Q ( − x, − τ ) = 0 , (5.2)respectively. Furthermore, by setting k = ε λ, e δ = εe β and taking limit ε → 0, the first type of one-solitonsolution (3.4) for the reverse-space discrete DNLS equation (1.3) converges to Q ( x, τ ) = 1 e − λx − iλ τ − β + λ ∗ σ ( λ − λ ∗ ) e − λ ∗ x − iλ ∗ τ + β ∗ , (5.3)with λ, β being complex parameters, which is one type of one-soliton soluiton for the reverse-space nonlocalDNLS equation (5.1). By setting k = ε λ, l = ε ω , and taking limit ε → 0, the second type of one-solitonsolution (3.11) for the reverse-space discrete DNLS equation (1.3) converges to Q ( x, τ ) = 1 (cid:113) σλ ( λ + ω ) e − λx − iλ τ − bi + (cid:113) − σω ( λ + ω ) e ωx − iω τ + di , (5.4)15ith λ, ω, b, d being real parameters, which is another type of one-soliton soluiton for the reverse-spacenonlocal DNLS equation (5.1). Setting k = ε λ, l = ε ω , and taking limit ε → 0, the one-soliton solution(4.4) for the reverse-space discrete DNLS equation (1.4) converges to Q ( x, τ ) = 1 (cid:113) σλ ( λ + ω ) e − λx − iλ τ + (cid:113) − σω ( λ + ω ) e ωx − iω τ , (5.5)with λ, ω being complex parameters, which is one-soliton soluiton for the reverse-space-time nonlocal DNLSequation (5.2). The N-soliton solution expressed in terms of Grammian and Casorati determinant solutionsfor two types of nonlocal discrete DNLS (1.3) and (1.4) via the bilinearisation-reduction approach are underinvestigation. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant nos. 11601247,11965014 and 11605096). Appendix A Constrains on the parameters in three-soliton solution for thereverse-space-time discrete DNLS equation (1.4) Applying the cross multiplication on three-soliton solution (2.28-2.29) with nonlocal reduction r n ( t ) = σq − n ( − t ), we obtain the following 126 constraints on the parameters: e δ λ = − σ ˆ B µ,ν K = − σM µ,ν,m,p ˜ A m,p e δ λ,j , e α λ = − σ ˜ A µ,ν J = − σN m,p,µ,ν ˜ B m,p e α j,λ ,λ, j ∈ { , , } ; µ, ν ∈ { , , }\{ λ } , ν > µ ; m, p ∈ { , , }\{ j } , p > m, (A.1) e δ λ = − σB µ,m,p e α ν,j A ν,λ,j N λ,µ,m,p , e α λ = − σA m,p,µ e δ j,ν B j,ν,λ M m,p,λ,µ ,λ, j ∈ { , , } ; µ, ν ∈ { , , }\{ λ } ; m, p ∈ { , , }\{ j } , p > m, (A.2) e δ e δ A , ,λ N , ,µ,ν + e δ e δ A , ,λ N , ,µ,ν + e δ e δ A , ,λ N , ,µ,ν + σ e δ e α ,λ B ,µ,ν + σ e δ e α ,λ B ,µ,ν + σ e δ e α ,λ B ,µ,ν = 0 ,e α e α B λ, , M µ,ν, , + e α e α B λ, , M µ,ν, , + e α e α B λ, , M µ,ν, , + σ e α e δ λ, A µ,ν, + σ e α e δ λ, A µ,ν, + σ e α e δ λ, A µ,ν, = 0 , λ ∈ { , , } ; µ, ν ∈ { , , }\{ λ } , ν > µ, (A.3) B λ,m,p + σ e δ ν N λ,ν,m,p + σ e δ µ N λ,µ,m,p + e δ µ e α j e α µ,j ˜ B λ,µ + e δ ν e α j e α ν,j ˜ B λ,ν + σ e δ µ e δ ν e α j A µ,ν,j K = 0 ,A m,p,λ + σ e α ν M m,p,λ,ν + σ e α µ M m,p,λ,µ + e α µ e δ j e δ j,µ ˜ A λ,µ + e α ν e δ j e δ j,ν ˜ A λ,ν + σ e α µ e α ν e δ j B j,µ,ν J = 0 , λ, j ∈ { , , } ; µ, ν ∈ { , , }\{ λ } , ν > µ ; m, p ∈ { , , }\{ j } , p > m, (A.4) σ e δ m e δ p e α κ ˜ A µ,ν N m,p,κ,λ + σ e δ m e δ m,λ A j,m,β + σ e δ p e δ p,λ A j,p,β + e δ m e α κ B m,κ,λ M j,m,µ,ν + e δ p e α κ B p,κ,λ M j,p,µ,ν + e α j,β = 0 ,σ e α m e α p e δ κ ˜ B µ,ν M κ,λ,m,p + σ e α m e α λ,m B β,j,m + σ e α p e α λ,p B β,j,p + e α m e δ κ A κ,λ,m N µ,ν,j,m + e α p e δ κ A κ,λ,p M µ,ν,j,p + e δ β,j = 0 , λ, j ∈ { , , } ; µ, ν ∈ { , , }\{ λ } , ν > µ ; β ∈ { , , }\{ λ } ; κ ∈ { , , }\{ β, λ } ; m, p ∈ { , , }\{ j } , p > m, (A.5)16 e δ e δ e α µ A , ,µ N , ,µ,λ + σ e δ e δ e α µ A , ,µ N , ,µ,λ + σ e δ e δ e α ν A , ,ν N , ,ν,λ + σ e δ e δ e α ν A , ,ν N , ,ν,λ + σ e δ e δ e α µ A , ,µ N , ,µ,λ + σ e δ e δ e α ν A , ,ν N , ,ν,λ + e δ e α µ e α ,µ B ,µ,λ + e δ e α ν e α ,ν B ,ν,λ + e δ e α µ e α ,µ B ,µ,λ + e δ e α ν e α ,ν B ,ν,λ + e δ e α µ e α ,µ B ,µ,λ + e δ e α ν e α ,ν B ,ν,λ + e δ e δ e α µ e α ν ˜ B , M , ,µ,ν + e δ e δ e α µ e α ν ˜ B , M , ,µ,ν + e δ e δ e α µ e α ν ˜ B , M , ,µ,ν + σ e δ e δ ,λ + σ e δ e δ ,λ + σ e δ e δ ,λ + σ e δ e δ e δ e α µ e α ν ˜ A µ,ν K +1 = 0 ,σ e α e α e δ µ B µ, , M µ,λ, , + σ e α e α e δ µ B µ, , M µ,λ, , + σ e α e α e δ ν B ν, , M ν,λ, , + σ e α e α e δ ν B ν, , M ν,λ, , + σ e α e α e δ µ B µ, , M µ,λ, , + σ e α e α e δ ν B ν, , M ν,λ, , + e α e δ µ e δ µ, A µ,λ, + e α e δ ν e δ ν, A ν,λ, + e α e δ µ e δ µ, A µ,λ, + e α e δ ν e δ ν, A ν,λ, + e α e δ µ e δ µ, A µ,λ, + e α e δ ν e δ ν, A ν,λ, + e α e α e δ µ e δ ν ˜ A , N µ,ν, , + e α e α e δ µ e δ ν ˜ A , N µ,ν, , + e α e α e δ µ e δ ν ˜ A , N µ,ν, , + σ e α e α λ, + σ e α e α λ, + σ e α e α λ, + σ e α e α e α e δ µ e δ ν ˜ B µ,ν J +1 = 0 , λ ∈ { , , } ; µ, ν ∈ { , , }\{ λ } , ν > µ. (A.6) References [1] M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear schr¨odinger equation, Phys. Rev.Lett. 110 (2013) 064105.[2] M.J. Ablowitz and Z.H. Musslimani, Integrable discrete PT symmetric model, Phys. Rev. E 90 (2014)032912.[3] M.J. Ablowitz and Z.H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math. J. Math. Phys. 59 (2018) 011501.[5] J. Yang, Physically significant nonlocal nonlinear Schr¨odinger equation and its soliton solutions, Phys.Rev. E 98 (2018) 042202.[6] X.Y. Wen, Z.Y. Yan and Y. Yang, Dynamics of higher-order rational solitons for the nonlocal nonlinearSchr¨odinger equation with the self-induced parity-time-symmetric potential, Chaos 26 (2016) 063123.[7] L.Y. Ma, S.F. Shen and Z.N. Zhu, Soliton solution and gauge equivalence for an integrable nonlocalcomplex modified Korteweg-de Vries equation, J. Math. Phys. 58 (2017) 103501.[8] B. Yang and J. Yang, Rogue waves in the nonlocal PT-symmetric nonlinear Schr¨odinger equation, Lett.Math. Phys. 109 (2019) 945-973.[9] B.F. Feng, X.D. Luo, M.J. Ablowitz and Z.H. Musslimani, General soliton solution to a nonlocal nonlin-ear Schr¨odinger equation with zero and nonzero boundary conditions, Nonlinearity 31 (2018) 5385-5409.[10] X. Deng, S.Y. Lou and D.J. Zhang, Bilinearisation-reduction approach to the nonlocal discrete nonlinearSchr¨odinger equations, Appl. Math. Comput. 332 (2018) 477-483.[11] Z. Xu and K. Chow, Breathers and rogue waves for a third order nonlocal partial differential equationby a bilinear transformation, Appl. Math. Lett. 56 (2016) 72-77.1712] M. G¨urses and A. Pekcan, Nonlocal nonlinear Schr¨odinger equations and their soliton solutions, J.Math. Phys. 59 (2018) 051501.[13] M. G¨urses and A. Pekcan, Nonlocal modified KdV equations and their soliton solutions by HirotaMethod, Comm. Nonlinear Sci. Numer. Simul. 67 (2019) 427-448.[14] L.Y. Ma and Z.N. Zhu, N-soliton solution for an integrable nonlocal discrete focusing nonlinearschr¨odinger equation, Appl. Math. Lett. 59 (2016) 115-121.[15] B. Yang and J. Yang, On general rogue waves in the parity-time-symmetric nonlinear Schr¨odingerequation, J. Math. Anal. Appl. 487 (2020) 124023.[16] J.L. Ji and Z.N. Zhu, Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equationthrough inverse scattering transform, J. Math. Anal. Appl. 453 (2017) 973-984.[17] J. Yang, General N-solitons and their dynamics in several nonlocal nonlinear Schr¨odinger equations, Phys. Lett. A 383 (2019) 328-337.[18] T. Tsuchida, Integrable discretizations of derivative nonlinear Schr¨odinger equations, J. Phys. A: Math.Gen. 35 (2002) 7827-7847.[19] R. Hirota,