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

Wave Motion 93 (2020) 102487

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Gram determinant solutions to nonlocal integrable discrete nonlinear Schrödinger equations via the pair reduction ∗

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

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

article

info

a b s t r a c t In this paper, Gram determinant solutions of local and nonlocal integrable discrete nonlinear Schrödinger (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. © 2019 Elsevier B.V. All rights reserved.

Article history: Received 27 January 2019 Received in revised form 26 November 2019 Accepted 29 November 2019 Available online 5 December 2019 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 Schrödinger (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

υn+1 =

(

z

φn (1)

ψn

z −1

)

υn , υn,t =

(

iψn φn−1 − 2i (z − z −1 )2 i(z −1 φn − z φn−1 )

) −i(z ψn − z −1 ψn−1 ) υ , −iφn ψn−1 + 2i (z − z −1 )2 n

(1)

(2)

where υn = (υn , υn )T 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 − ψn φn (ψn+1 + ψn−1 ), ∗ Correspondence to: Department of Mathematics, Lishui University, Lishui, 323000, China. E-mail address: [email protected] (J. Chen). https://doi.org/10.1016/j.wavemoti.2019.102487 0165-2125/© 2019 Elsevier B.V. All rights reserved.

(2)

2

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

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

(3)

It admits four different symmetry reductions [3,12,13,19,22,24]: (i) The standard Ablowitz–Ladik symmetry (φn = δψn∗ ) gives rise to the local IDNLS equation iψn,t = ψn+1 + ψn−1 − 2ψn − δψn ψn∗ (ψn+1 + ψn−1 ), δ = ±1.

(4)

∗ (ii) The discrete PT preserved symmetry (φn = δψ− n ) leads to the PT symmetric IDNLS equation ∗ iψn,t = ψn+1 + ψn−1 − 2ψn − δψn ψ− n (ψn+1 + ψn−1 ), δ = ±1.

(5)

(iii) The reverse time symmetry (φn = γ ψn (−t)) yields the reverse time symmetric IDNLS equation iψn,t = ψn+1 + ψn−1 − 2ψn − γ ψn ψn (−t)(ψn+1 + ψn−1 ),

(6)

where γ is an arbitrary complex constant. (iv) The reverse discrete-time symmetry (φn = γ ψ−n (−t)) results in the reverse discrete-time symmetric IDNLS equation iψn,t = ψn+1 + ψn−1 − 2ψn − γ ψn ψ−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 soliton 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 1 Eq. (4) was realized by imposing the constraint condition qi = p∗− [27,28] on the before-reduction IDNLS equation. i 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-reduction 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-reduction 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-reduction) 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. 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 Bäcklund 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

⏐ (n) ⏐ϕ (k, l) ⏐ 1 ⏐ (n) ⏐ϕ (k, l) τn (k, l) = ⏐⏐ 2 . ⏐ .. ⏐ (n) ⏐ϕ (k, l) N

ϕ1(n+1) (k, l)

···

ϕ2(n+1) (k, l) .. .

···

ϕN(n+1) (k, l)

··· ···

⏐ ϕ1(n+N −1) (k, l)⏐⏐ ⏐ ϕ2(n+N −1) (k, l)⏐⏐ .. ⏐, ⏐ . ⏐ (n+N −1) ϕN (k, l)⏐

(11)

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

3

(n)

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

∂x ϕi(n) (k, l) = ϕi(n+1) (k, l), ∂y ϕi(n) (k, l) = ϕi(n−1) (k, l), (n)

(n+1)

∆k ϕi (k, l) = ϕi

(n)

(n−1)

(k, l), ∆l ϕi (k, l) = ϕi

(12)

(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, l − 1) , ∆l f (k, l) = , a a and the Hirota’s bilinear operators Dx and Dy are defined as

∆k f (k, l) =

f (k, l) − f (k − 1, l)

Dnx Dm y (a · b) =

(

∂ ∂ − ′ ∂x ∂x

)n (

∂ ∂ − ′ ∂y ∂y

)m

⏐ ⏐

a(x, y)b(x′ , y′ )⏐⏐

x=x′ ,y=y′

(14)

.

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 following, the (n) function ϕi (k, l) is taken as

)−l ( )−l ( 1 1 exp(ξi ) + qni (1 − aqi )−k 1 − b exp(ηi ), ϕi(n) (k, l) = pni (1 − api )−k 1 − b pi

qi

(15)

with

ξ i = pi x +

1 pi

y + ξi,0 , ηi = qi x +

1

y + ηi,0 ,

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, one need to impose the constraint condition for the wave numbers in (15) api qi + b − pi − qi = 0,

(16)

which implies p2i

1 − b p1

i

1 − api

= q2i

1 − b q1

i

1 − aqi

.

(17)

(n)

This condition makes ϕi (k, l) in (15) satisfy

ϕi(n+2) (k + 1, l − 1) = p2i

1 − b p1

i

1 − api

ϕi(n) (k, l),

(18)

and then tau functions have the relations

( τn+2 (k + 1, l − 1) =

N ∏ i=1

p2i

1 − b p1

)

i

1 − api

τn (k, l).

(19)

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 are converted to the bilinear equations 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)

fn+1 fn−1 − fn2 = (ab − 1)(fn2 − gn hn ),

(27)

4

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

which have the solution in terms of Casorati determinant as follows

⏐ ⏐

n+j−1 ξi e

fn = |F | = ⏐pi

⏐ ⏐ + qni +j−1 eηi ⏐ ,

(28)

⏐ ⏐ n+j ⏐ ⏐ pn+j qi ⏐ i ξi ηi ⏐ e + e ⏐, gn = |G| = ⏐ ⏐ 1 − api 1 − aqi ⏐ ⏐ ⏐ ⏐ n+j−2 ξi n+j−2 ηi ⏐ hn = |H | = ⏐(1 − api )pi e + (1 − aqi )qi e ⏐,

(29) (30)

with

) ( ) ( bd bd t + ξi,0 , ηi = i acqi + t + ηi,0 ξi = i acpi + pi

(31)

qi

where the wave numbers pi and qi need to satisfy the constraint condition (16). Through the dependent variable transformation un =

gn fn

hn

, vn =

(32)

fn

one can get the generalized IDNLS equation dun

+ (c + d)un − [ab − (ab − 1)un vn ](dun+1 + cun−1 ) = 0, dt dvn −i + (c + d)vn − [ab − (ab − 1)un vn ](c vn+1 + dvn−1 ) = 0. dt i

(33) (34)

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(ξi ) = ⎣ (qk − pi )⎦ exp(ξi′ ), exp(ηi ) = ⎣ (qk − qi )⎦ exp(ηi′ ), ⎡

(35)

k=1 k̸ =i

k=1 k̸ =i

with

(

ξi = i acpi + ′

bd

) t+

pi

ξi′,0 ,

(

ηi = i acqi + ′

bd

)

qi

t + ηi′,0 ,

and introduce the following 2N × 2N diagonal matrices A = Diag(a1 , a2 . . . , a2N ), ai = B = Diag(b1 , b2 . . . , b2N ), bi = C = Diag(c1 , c2 , . . . , c2N ), ci =

1 qi − pi 1

,

,

pni eξi 1 − api ′

(36)

pi

(37)

,

(38)

and a Vandermonde matrix

⎡

⎤ (2N −i ∏

∑

⎢

2N −i Vq = ⎢ ⎣(−1)

1≤k1
l=1

) .

⎥

qkl ⎥ ⎦

(39)

2N ×2N

For example, when N = 1 and N = 2, the Vandermonde matrices read

( Vq =

−q2

−q1

1

1

⎛

)

,

−q2 q3 q4 ⎜q2 q3 + q2 q4 + q3 q4 Vq = ⎝ −q2 − q3 − q4 1

(40)

−q1 q3 q4 q1 q3 + q1 q4 + q3 q4 −q1 − q3 − q4 1

−q1 q2 q4 q1 q2 + q1 q4 + q2 q4 −q1 − q2 − q4 1

⎞

−q1 q2 q3 q1 q2 + q1 q3 + q2 q3 ⎟ . −q1 − q2 − q3 ⎠ 1

(41)

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

5

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

⏐ ⏐ ⏐ n ηi′ ⏐ ˜fn = |F˜ | = |ABFVq | = ⏐⏐ 1 + δij 1 qi e ′ ⏐⏐ , ⏐ pi − qj pi − qj pn eξj ⏐ j 2N ×2N ⏐ ⏐ [ ] ′ ⏐ ⏐ 1 qni eηi ⏐ qi (1 − api ) 1 ⏐ ˜ | = |CABFVq | = ⏐ + δij g˜n = |G , ′⏐ ⏐ pi − qj pj (1 − aqj ) pi − qj pn eξj ⏐ j 2N ×2N ⏐ ⏐ [ ] ⏐ 1 n ηi′ ⏐ q e p (1 − aq ) 1 ⏐ i i i ˜hn = |H˜ | = |C −1 ABFVq | = ⏐⏐ + δij , ′⏐ ⏐ pi − qj qj (1 − apj ) pi − qj pn eξj ⏐ j

(42)

(43)

(44)

2N ×2N

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

⏐

⏐

⏐A f˜n = |F˜ | = |PABFVq Q | = ⏐⏐ C

B ⏐⏐ D⏐

, ⏐ B ⏐⏐ D(g) ⏐

(47)

2N ×2N

⏐

⏐ ˜ | = |CPABFVq Q | = ⏐A(g) g˜n = |G ⏐ C

,

(48)

2N ×2N

⏐ B ⏐⏐ D(h) ⏐

⏐ ⏐ ˜hn = |H˜ | = |C −1 PABFVq Q | = ⏐A(h) ⏐ C

,

(49)

2N ×2N

with the block matrices given by

( ) ( ) , A(g) = aij (1) N ×N , A(h) = aij (−1) N ×N , ( ) ( ) ( ) D = dij (0) N ×N , D(g) = dij (1) N ×N , D(h) = dij (−1) N ×N , ( ) ( ) B = bij N ×N , C = cij N ×N , (

A = aij (0)

)

(50)

N ×N

(51) (52)

where the elements are defined by aij (k) =

cij =

1 pi − qj pN +i

pN +i − qj

+ δij

[

1 pi − qj

, dij (k) =

qi (1 − api )

]k

pN +i − qN +j

′

′ n ξj

pj (1 − aqj ) pN +i qN +j

qni eηi pj e

+ δij

, bij =

pN +i qN +j

qN +j pi − qN +j

[

pN +i − qN +j

,

qN +i (1 − apN +i )

(53)

]k

pN +j (1 − aqN +j )

qnN +i e

′ ηN +i

pnN +j e

ξN′ +j

.

(54)

This Gram determinant solution also satisfies 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 pN +i =

1 q∗i

, qN +i =

1 p∗i

, b = a∗ ,

(55)

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 d = c ∗ , ξN′ +i,0 = −ηi′∗,0 , ηN′ +i,0 = −ξi′∗ ,0 ,

(56)

then the following relations hold n −ηi n −ξi ξN′ +i = −ηi′∗ , ηN′ +i = −ξi′∗ , pnN +i eξN +i = q∗− e , qnN +i eηN +i = p∗− e . i i ′

′∗

′

′∗

(57)

The elements in the Gram determinant solution (47)–(49) are rewritten as aij (k) =

1 pi − qj

+ δij

1 pi − qj

[

qi (1 − api ) pi (1 − aqi )

]k

qni eηi

′

′ n ξj

pj e

, bij =

1 ∗

pi pj − 1

,

(58)

6

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

1

cij =

1 − q∗i qj

1

, dij (k) =

+ δij

p∗j − q∗i

[

1 p∗j − q∗i

a − q∗i

q∗j n e

]k

ηj′∗

.

p∗i n eξi

′∗

a − p∗i

(59)

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

˜ T T | = |F˜ | = h˜ n , f˜n∗ = |T F˜ T T | = |F˜ | = f˜n , g˜n∗ = |T H

(60)

where the matrix T is defined by

( T =

0

IN × N 0

IN × N

)

.

(61)

2N ×2N g˜n∗ f˜ ∗

From the transformation (32), it immediately follows u∗n = transformations

) ( eiθ + e−iθ ψn t − 2it , exp −inθ + i α |a|2

α=

un = √

=

n

h˜ n f˜n

= vn . Furthermore, through the variable

|a|2 − 1 e−iθ , c= , 2 δ|a| |a|2

(62)

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

√ ψn =

(

|a|2 − 1 e δ|a|2

iθ −iθ inθ −i e +e2 t +2it |a|

)

g˜n f˜n

,

(63)

with δ = 1(|a|2 > 1) and δ = −1(|a|2 < 1), where tau functions are given by

⏐

⏐A f˜n = ⏐⏐ C

⏐

B ⏐⏐ D⏐

2N ×2N

⏐

⏐ ⏐A , g˜n = ⏐⏐ (g) C

B ⏐⏐ D(g) ⏐

,

(64)

2N ×2N

with the block matrices

(

1

A=

+ δij

pi − qj

(

1

D=

′

(

) , A(g) =

pi − qj pn eξj j

+ δij

p∗j − q∗i

qni eηi

1

′

q∗j n e

1

1

(

)

j

p∗j − q∗i p∗n eξi i

, D(g) =

′∗

+ δij

pi − qj

N ×N η′∗

N ×N

1 p∗j − q∗i

[

1 pi − qj

+ δij

qi (1 − api )

]

pi (1 − aqi ) 1

p∗j − q∗i

[

a − q∗i a − p∗i

qni eηi

′

) ,

ξj′

pnj e

]

q∗j n e

N ×N η′∗ j

p∗i n eξi

)

,

′∗

N ×N

and

( B=

)

1

, C=

pi p∗j − 1

(

Here ξi′ = i

ae−iθ |a|2

N ×N a∗ eiθ |a|2 pi

pi +

)

(

)

1 1 − q∗i qj

. N ×N

t + ξi′,0 and ηi′ = i

(

ae−iθ |a|2

qi +

a∗ eiθ |a |2 q i

)

t + ηi′,0 , pi , qi ξi′,0 , ηi′,0 (i = 1, 2 . . . , N) and a are arbitrary

complex constants, θ is an arbitrary real constant, these parameters need to satisfy the constraint conditions api qi + 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. For example, by taking N = 1, tau functions for the local IDNLS equation (4) are written as:

( fn = ∆1 + ∆2

qn1 eη1 ′

1+

pn1 eξ1 ′

( gn = ∆1 + ∆2

q∗1n eη1

′∗

+

qn1 eη1

p∗1n eξ1

′∗

′

1 + P1

pn1 eξ1 ′

+

|q1 |2n eη1 +η1 ′

+

|p1 |2n eξ1 +ξ1

′∗ q∗n eη1 P1∗ 1 ′∗ ∗n ξ1

p1 e

′∗

′

′∗

) ,

(66)

|q1 |2n eη1 +η1 ′

2

+ |P1 |

′∗

)

|p1 |2n eξ1 +ξ1 ′

′∗

,

(67)

where 1

(|p1 |2 − 1)(|q1 |2 − 1)

(

)

, ∆2 =

1

, P1 = 2

a∗ − p 1

, a∗ − q1 |p1 − q1 | ( ) −iθ ∗ iθ −iθ ∗ iθ and ξ1′ = i ae 2 p1 + a 2e t + ξ1′ ,0 and η1′ = i ae 2 q1 + a 2e t + η1′ ,0 , p1 , q1 ξ1′ ,0 , η1′ ,0 and a are arbitrary complex |a| |a| p1 |a | |a| q1 constants, θ is an arbitrary real constant, they need to satisfy the constraint condition ap1 q1 + a∗ − p1 − q1 = 0. ∆1 =

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

7

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). To this end, we first impose the pair conditions pN +i = q∗i , qN +i = p∗i , 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 c ∗ = c , d∗ = d, ξN′ +i,0 = −ηi′∗,0 , ηN′ +i,0 = −ξi′∗ ,0 ,

(69)

one can find the relations

ξN′ +i = −ηi′∗ , ηN′ +i = −ξi′∗ , pnN +i eξN +i = q∗i n e−ηi , qnN +i eηN +i = p∗i n e−ξi . ′

′

′∗

′∗

(70)

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

⏐ ⏐ ˜fn = |F˜ | = ⏐A ⏐

⏐

B ⏐⏐ D⏐

C

2N ×2N

⏐ ⏐A ˜ , g˜n = |G| = ⏐⏐ (g) C

⏐

B ⏐⏐ D(g) ⏐

2N ×2N

⏐ ⏐A ˜ ˜ , hn = |H | = ⏐⏐ (h) C

⏐

B ⏐⏐ D(h) ⏐

,

(71)

2N ×2N

with the block matrices given by (50)–(52) whose elements are defined by aij (k) =

1 pi − qj 1

cij =

+ δij

pi − qj

, dij (k) =

q∗i − qj

[

1

qni eηi

′

]k

qi (1 − api )

ξ′ pnj e j

pj (1 − aqj ) 1

q∗i − p∗j

+ δij

[

1 q∗i − p∗j

1

, bij =

pi − p∗j

p∗i (1 − aq∗i )

,

(72)

p∗i n eηi

′∗

]k

ξj′∗

q∗i (1 − ap∗i )

q∗j n e

.

(73)

T T , B∗ = −BT and C ∗ = −C T , thus one can , A∗ (−n)(h) = −D(g) In this situation, we have A∗ (−n) = −DT , A∗ (−n)(g) = −D(h) get

∗ ˜T ˜ ˜ f˜−∗n = |−T F˜ T T | = |F˜ | = f˜n , g˜− n = |−T H T | = |F | = hn .

(74)

From the transformation (32), it immediately reaches u∗−n = transformations

∗ g˜− n ∗ f˜− n

=

h˜ n f˜n

= vn . Furthermore, through the variable

) ( ψn ab − 1 e−θ eθ eθ + e−θ t − 2it , α = , c= , d= , exp −nθ + i ab δ ab ab ab α

un = √

(75)

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

√ ψn =

ab − 1

δ ab

(

e

θ −θ nθ −i e +abe t +2it

)

g˜n f˜n

,

(76)

with δ = 1(ab > 1, or ab < 0) and δ = −1(0 < ab < 1), where tau functions are given by

⏐

⏐A f˜n = ⏐⏐ C

⏐

B ⏐⏐ D⏐

2N ×2N

⏐ ⏐A , g˜n = ⏐⏐ (g) C

⏐

B ⏐⏐ D(g) ⏐

,

(77)

2N ×2N

with the block matrices

( A=

1

+ δij

pi − qj

( D=

1 q∗i − p∗j

qni eηi

′

1

)

( , A(g) =

pi − qj pn eξj j

+ δij

′

1 q∗i −

N ×N ∗n ηi′∗ pi e ′∗ p∗j q∗n eξj j N ×N

B=

1 pi − p∗j

) , C= N ×N

pi − qj

)

( , D(g) =

and

(

1

(

1 q∗i − qj

)

. N ×N

+ δij 1

q∗i − p∗j

[

1 pi − qj

+ δij

qi (1 − api ) pj (1 − aqj )

1 q∗i − p∗j

[

]

qni eηi

′

) ,

ξj′

pnj e

p∗i (1 − aq∗i ) q∗i (1 − ap∗i )

N ×N

]

p∗i n eηi

′∗

q∗j n e

) ,

ξj′∗ N ×N

8

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

Here ξi′ = i

(

e−θ b

eθ api

pi +

)

t + ξi′,0 and ηi′ = i

(

e−θ b

qi +

eθ aqi

)

t + ηi′,0 , pi , qi ξi′,0 and ηi′,0 (i = 1, 2 . . . , N) are arbitrary complex

constants, a, b and θ are arbitrary real constants, these parameters need to satisfy the constraint conditions api qi + b − pi − qi = 0.

(78)

For example, tau functions for the nonlocal IDNLS equation (5) are written as:

( fn = ∆1 + ∆2

qn1 eη1

1+

gn = ∆1 + ∆2

+

pn1 eξ1 ′

(

p∗1n eη1

′∗

′

qn1 eη1 pn1 eξ1 ′

′

+

q∗1n eξ1

′∗

′

′∗

+

′∗

) ,

pn1 q∗1n eξ1 +ξ1

1 p∗1n eη1

′

1 + P1

qn1 p∗1n eη1 +η1

P1∗ q∗n eξ1 1

′∗

′∗

(79) ′

+

)

P1 qn1 p∗1n eη1 +η1

′∗

(80)

P1∗ pn q∗n eξ1 +ξ1 1 1 ′

′∗

where

∆1 = and ξ1′ = i

(

1 ∗

∗

(p1 − p1 )(q1 − q1 ) e−θ b

p1 +

eθ ap1

)

1

, ∆2 = −

2

|p1 − q1 | ( −θ and η1′ = i e b q1 +

t + ξ1′ ,0

q1 (1 − ap1 )

, P1 = eθ aq1

)

p1 (1 − aq1 )

,

t + η1′ ,0 , p1 , q1 ξ1′ ,0 and η1′ ,0 are arbitrary complex constants, a,

b and θ are arbitrary real constants, they need to satisfy the constraint condition ap1 q1 + b − p1 − q1 = 0. To investigate the asymptotic behaviour of above two-soliton solution (79)–(80), we redefine the following notations:

θˆ = nθ − i

eθ + e−θ ab

√ t + 2it , ρˆ =

ab − 1

δ ab qn1 eη1

,

′ pn1 eξ1

p1

∆2 = δˆ ec1 , ∆1 + ∆2

= eρ+iϕ , P1 = ec0 +iθ1 ,

p∗1n eη1

(81)

′∗

′

η1′ − ξ1′ = (A + iB)t + A0 + iB0 ,

q1

= eχ1 ,

′∗ q∗1n eξ1

= eχ2 .

(82)

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 ≈ −∞): +

ψn → ρˆ e

θˆ

1 + δˆ eX1

+c1 +c0 i(Y1+ +θ1 )

e

+

1 + δˆ eX1

+c1 iY1+

e

⏐ ⏐ ⏐ ab − 1 ⏐ 2nθ +c cosh(X1+ + c1 + c0 ) + δˆ cos(Y1+ + θ1 ) 0 ⏐ ⏐e , |ψn | → ⏐ . δ ab ⏐ cosh(X1+ + c1 ) + δˆ cos(Y1+ ) 2

(83)

Soliton 2 (χ2 ≈ 0, χ1 ≈ −∞): −

ψn → ρˆ eθ ˆ

1 + δˆ e−X1

+c1 −c0 i(Y1− +θ1 )

e

−

1 + δˆ e−X1

+c1 iY1−

e

⏐ ⏐ ⏐ ab − 1 ⏐ 2nθ −c cosh(X1− − c1 + c0 ) + δˆ cos(Y1− + θ1 ) 0 ⏐e , |ψn |2 → ⏐⏐ . δ ab ⏐ cosh(X1− − c1 ) + δˆ cos(Y1− )

(84)

⏐ ⏐ ⏐ ab − 1 ⏐ 2nθ −c cosh(X1+ + c0 ) + cos(Y1− + θ1 ) 0 ⏐e , |ψn |2 → ⏐⏐ . δ ab ⏐ cosh(X1+ ) + δˆ cos(Y1+ )

(85)

(ii) After collision (t → +∞) Soliton 1 (χ1 ≈ 0, χ2 ≈ +∞): +

ψn → ρˆ eθ ˆ

p1 1 + e−X1

−c0 −i(Y1+ +θ1 )

e

+

+

p∗1

1 + e−X1 e−iY1

Soliton 2 (χ2 ≈ 0, χ1 ≈ +∞): −

ψn → ρˆ eθ ˆ

p1 1 + eX1

+c0 −i(Y1− +θ1 )

e

−

p∗1

−

1 + eX1 e−iY1

⏐ ⏐ ⏐ ab − 1 ⏐ 2nθ +c cosh(X1− + c0 ) + cos(Y1− + θ1 ) 0 ⏐e , |ψn |2 → ⏐⏐ . δ ab ⏐ cosh(X1− ) + δˆ cos(Y1− )

(86)

Here X1± = nρ ± At ± A0 and Y1± = nϕ ± Bt ± B0 . In the previous work [27], one of the authors has applied a direct reduction qi = p∗i to derive the solution of the nonlocal IDNLS equation (5). When N = 1, the solution reads

√ ψn =

ab − 1

δ ab

(

e

θ −θ t +2it nθ−i e +abe

)

[

∗

p1 (1 − ap1 )

] 1+

p1 (1 − ap∗1 )

where, δ = 1(ab < 0) and δ = −1(0 < ab < 1), ξi′ = i

p1 (1−ap∗ ) 1 p∗ (1−ap ) 1

1+

(

pi beθ

+

( )n p1 p∗ 1

1

( )n p1 p∗ 1

eθ api

)

′∗

eξ1 +ξ1 ′

′∗

eξ1 +ξ1 ′

,

t + ξi′,0 , p1 and ξ1′ ,0 are arbitrary complex parameters,

a, b and θ are arbitrary real parameters and they need to satisfy 1 − ap1 = − p∗1 (1 −

[

(1−ap∗ ) 1

]

(87)

p

1

b ) p1

and ab < 1. For the solution (87),

= e2iγ1 and define ξ1 + ξ1∗ ≡ 2ϑ1 , then the modular square of ψn is given by ⏐ ⏐[ ] ⏐ 1 − ab ⏐ ⏐ 1 − 2 sin(2nβ1 + β1 + γ ) sin(β1 + γ ) , |ψn |2 = exp(2nθ ) ⏐⏐ ab ⏐ cosh(2ϑ1 ) + cos(2nβ1 )

if we set pi = exp(αi + iβi ),

(1−ap1 )

(88)

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

9

Fig. 1. The solution (79)–(80) of the PT symmetric IDNLS equation (5) with the parameters (a) a = 1, 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 =

3 , 2

18 , 5

p1 = 3 + 3i and q1 =

p1 = 2 + 3i and q1 =

19 20

+

3 5

+ 53 i; (b)

3 i. 20

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 the nonsingular solution. We illustrate the solution (79)–(80) in Fig. 1 with the parameters ξ1′ ,0 = η1′ ,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, Fig. 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 specifically, by imposing the pair conditions pN +i =

1 qi

, qN +i =

1 pi

, b = a,

(89)

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 d = c , ξN′ +i,0 = −ηi′,0 , ηN′ +i,0 = −ξi′,0 ,

(90)

then the following relations hold n −ηi (−t) n −ξi (−t) ξN′ +i = −ηi′ (−t), ηN′ +i = −ξi′ (−t), pnN +i eξN +i = q− , qnN +i eηN +i = p− . i e i e ′

′

′

′

(91)

10

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

The elements in the Gram determinant solution (47)–(49) are rewritten as aij (k) =

1 pi − qj 1

cij =

+ δij

1 − qi qj

[

1

qi (1 − api )

pi − qj

, dij (k) =

qni eηi

′

]k

ξ′ pnj e j

pi (1 − aqi ) 1

pj − qi

+ δij

[

1 pj − qi

, bij =

pi pj − 1

qnj e

]k

a − qi

1

(92)

ηj′ (−t)

pni eξi (−t) ′

a − pi

,

.

(93)

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

˜ T T | = |F˜ | = h˜ n . f˜n (−t) = |T F˜ T T | = |F˜ | = f˜n , g˜n (−t) = |T H

(94)

From the transformation (32), it immediately reaches un (−t) = transformations

g˜n (−t) f˜n (−t)

=

h˜ n f˜n

= vn . Furthermore, through the variable

[ ( ) ] 1 a2 − 1 1 ψn exp 2i − 1 t , α= , c = 2, 2 2 a γa a α

(95)

un = √

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

√ ψn =

a2 − 1

γ a2

e

( −2i

1 a2

) ˜n −1 t g

f˜n

,

(96)

where tau functions are given by

⏐

⏐A f˜n = ⏐⏐ C

⏐

B ⏐⏐ D⏐

2N ×2N

⏐

⏐ ⏐A , g˜n = ⏐⏐ (g) C

B ⏐⏐ D(g) ⏐

,

(97)

2N ×2N

with the block matrices

(

1

A=

+ δij

pi − qj

( D=

1

+ δij

pj − qi

qni eηi

′

1

pi − qj pn eξj j

)

( , A(g) =

′

N ×N

ηj′ (−t)

qnj e

1

pj − qi pn eξi (−t) i

1 pi − qj

)

( , D(g) =

′

+ δij

1 pj − qi

N ×N

[

1 pi − qj

+ δij

qi (1 − api ) pi (1 − aqi )

1 pj − qi

[

a − qi a − pi

]

]

qni eηi

′

) ,

ξj′

pnj e qnj e

ηj′ (−t)

pni eξi (−t)

N ×N

) ,

′

N ×N

and

( B=

)

1 pi pj − 1

(

Here ξi′ = i pi +

1 pi

, C=

(

N ×N

)

t a

)

1 1 − qi qj

. N ×N

( + ξi′,0 and ηi′ = i qi +

1 qi

)

t a

+ ηi′,0 , pi , qi ξi′,0 , ηi′,0 (i = 1, 2 . . . , N) and a are arbitrary complex

constants, these parameters need to satisfy the constraint conditions api qi + a − pi − qi = 0.

(98)

For example, by taking N = 1, tau functions for the nonlocal IDNLS equation (6) are written as:

( fn = ∆1 + ∆2

qn1 eη1 ′

1+

pn1 eξ1

( gn = ∆1 + ∆2

′

qn1 eη1 (−t) ′

+

qn1 eη1

pn1 eξ1 (−t) ′

′

1 + P1

pn1 eξ1 ′

η1 +η1 (−t) q2n 1 e ′

+

) ,

ξ1 +ξ1 (−t) p2n 1 e ′

1 qn1 eη1 (−t) ′

+

′

P1 pn eξ1 (−t) 1 ′

′

η1 +η1 (−t) q2n 1 e ′

+

(99)

′

ξ1 +ξ1 (−t) p2n 1 e ′

′

) ,

(100)

where

∆1 =

(

1 (p21 − 1)(q21 − 1)

1 (p1 − q1 )2

, P1 =

a − p1 a − q1

,

( ) + ξ1′ ,0 and η1′ = i q1 + q1 at + η1′ ,0 , p1 , q1 ξ1′ ,0 , η1′ ,0 and a are arbitrary complex constants, they 1 need to satisfy the constraint condition ap1 q1 + a − p1 − q1 = 0.

and ξ1′ = i p1 +

1 p1

)

, ∆2 =

t a

J. Chen, B.-F. Feng and Y. Jin / Wave Motion 93 (2020) 102487

11

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 multi-soliton with the even number respectively. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements J.C. acknowledges support from National Natural Science Foundation of China (No. 11705077). B.F. is partially supported by Natural Science Foundation under Grant No. DMS-1715991 and National Natural Science Foundation of China under Grant No. 11728103. Y.J. is supported by National Natural Science Foundation of China under Grant Nos. 11771395 and 11571306. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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