Multi-rogue wave solutions for a generalized integrable discrete nonlinear Schrödinger equation with higher-order excitations

In this paper, we construct the discrete higher-order rogue wave (RW) solutions for a generalized integrable discrete nonlinear Schrödinger (NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation is constructed. Second, the dynamical behaviors of first-, second- and third-order RW solutions are investigated in corresponding to the unique spectral parameter, higher-order term coefficient, and free constants. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz–Ladik (AL) equation are illustrated in figures. Moreover, we explore the numerical experiments, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.


Introduction
Rogue wave was founded in many fields, such as nonlinear optics, fluid mechanics, and even finance [1][2][3]. A mass of nonlinear evolution equations including the NLS equation, Kundu-Eckhaus equation, Hirota equation, Sasa-Satuma equation, nonlinear wave equation and so on, can describe the RW phenomena [4][5][6][7][8][9][10]. As a basic model that describes optical soliton propagation in Kerr media, the NLS equation contains multi-soliton solutions, breather solutions, and RW solutions [5,[11][12][13]. However, in the regime of ultra-short pulses, the NLS equation is inappropriate to accurately describe the phenomena, and higher-order nonlinear dispersion terms must be taken into account [6][7][8][9]14]. In discrete integrable system, the RW solutions of the AL equation, coupled discrete NLS equation and discrete Hirota equation are also discussed based on generalized Darboux transformation (DT) and Hirota bilinear method [15][16][17][18]. There are great differences on RWs between the continuous integrable system and discrete integrable system. Ohta and Yang point out that the RWs can exist in the defocusing Ablowitz-Ladik equation [17].
As we know, the higher-order NLS equation named as the Lakshmanan-Porsezian-Daniel (LPD) equation [19] iq t + q x x + 2|q| 2 q + γ q x x x x + 8|q| 2 q x x + 6q * q 2 x + 4q|q x | 2 +2q 2 q * x x + 6|q| 4 q = 0. (1) is the third member of the NLS hierarchy. Here q is a varying wave packet envelope, q * denotes the complex conjugate q, and γ is a real parameter and stands for the strength of higher-order linear and nonlinear effects. Equation (1) can also describe the dynamics of higher-order alpha-helical proteins with nearest and next nearest neighbour interactions [20,21]. This equation has attracted great attentions. In Refs. [19,22], Authors establish the relation between higher-order NLS equation and one-dimensional Heisenberg ferromagnetic chains when higher order spin-spin exchange interactions (biquadratic type) and the effect of discreteness are considered. The integrability of Eq. (1) including its singularity structure, construction of Lax pair and Bäcklund transformation have been discussed in detail in Ref. [22]. The one soliton solution of Eq. (1) has been constructed [20] by using Hirota method. Multisoliton solutions using DT is presented in [23]. Rogue waves for the three-coupled fourth-order NLS system is studied in [24]. Besides, Eq. (1) can be regarded as a special case for an integrable three-parameter fifthorder nonlinear Schrödinger equation [25,26]. Rational solutions, breather solutions, rogue wave and modulation instability of this integrable three-parameter fifthorder nonlinear Schrödinger equation are analytically studied based on DT and robust inverse scattering transform [27,28]. The corresponding rational solutions and breather solutions of Eq. (1) can be obtained under certain constraints.
In this article, we focus on the following spatial discretization [20] of integrable higher-order NLS equation (1) Equation (2) can govern the discrete α-helical protein chain model with several higher-order excitations and interactions. Under the transformation the higher-order integrable discrete NLS equation (2) yields the integrable fourth-order NLS equation (1). Reference [20] investigates the integrability of Eq. (2) including Hamiltonian, discrete Lax pair, discrete soliton and gauge equivalence. However, as we know, there is little work on rogue wave solutions and breather solutions of this higher-order integrable discrete NLS equation (2). This is the main motivation for us to investigate the higher-order RWs of the discrete integrable NLS equation (2) with higher-order excitations in this paper. Moreover, it is very meaningful to study other integrable properties of the higher-order integrable discrete NLS equation (2). We shall give an insight into the continuous limit theory of higher-order integrable discrete NLS equation (2) including discrete DT, discrete rational solutions, discrete breather solutions and gauge equivalence in the future. The paper is organized as follows. In Sect. 2, by using the modified discrete Lax pairs, we apply the generalized (1,N-1)-fold Darboux transformation [5,15] to construct higher-order discrete RW solutions of Eq. (2). The dynamical behaviors of these discrete RWs are discussed in Sect. 3, which exhibits interesting wave structures. Finally, in Sect. 4 the modulation instability of continuous-wave states of the higher-order discrete NLS equation (2) is investigated.

Lax pair and generalized discrete DT
The higher-order discrete NLS equation (2) admits the following discrete modified Lax pair where the shift operator E is defined as Eϕ n = ϕ n+1 , ϕ n = (ϕ n,1 , ϕ n,2 ) T is the vector eigenfunction. The matrices U n and V n with spectral parameter λ take the forms in which One can directly verify that the discrete zero curvature condition U n,t = (E V n )U n − U n V n of the linear spectral equations (4) yields the generalized integrable discrete NLS equation (2).
Following the idea in [29], the Darboux transformation of the higher-order discrete NLS equation (2) can be obtained. Under the gauge transformation with can be determined by The linear spectral problem (4) changes to new one as and the matricesŨ n andŨ n satisfỹ The relation between potentialq n [N ] and potential q n is where with It is noted that the expression of 1 [N ] and 2 [N ] can be derived by substituting 2 ) T for the first and second column in [N ], respectively.
Next, we will construct the generalized (1, N − 1)fold DT for higher-order discrete NLS equation (2). The generalized (1, N − 1)-fold DT links to single spectral parameter λ = λ 1 and the order N-1 of the highest-order derivatives for the eigenfunctions. Using the similar method in Ref. [5,18], we get a generalized (1, N −1)-fold DT for higher-order discrete NLS equation (2). Especially, we consider the following eigenfunction solution of the Lax pair (4) with seed solution q 0 (n, t) = ce iφt with C j ( j = 1, 2) are arbitrary complex parameters (i.e.,C 1 = 1, C 2 = 0), d k , f k are free real and is small parameter. We fix the spectral parameter λ = λ 1 + 2 with λ 1 = √ 1 + c 2 ± c in Eq. (11) and expand eigenfunction ϕ(λ) into the Taylor series at = 0, then we obtain where Then we obtain a generalized DT for the higher-order discrete NLS equation (2) q where  2 are described by [N ] respectively, but the first row and the second row in the [N ] are changed to (λ N +1 φ 1 , . . . , φ 1

RW solutions and dynamic behaviors
Case 1: one-order RW solutions As N = 1, the solution (13) reduces with For convenient, choose h = 1, c = 3 4 corresponding to λ 1 = 1 2 then we obtain the first-order RW solution of Eq. (2) is Note that q n+n 0 [1] is a solution with arbitrary real number shift n 0 and the translational property also satisfies the following higher-order RW solutions. Next we illustrate the property of first-order RW solution (15).
By analyzing the explicit formula of q n+n 0 [1], we find that the parameter γ produces no effect in the amplitude of the first-order RW solution (15). The maximum amplitude of |q n [1]| is 63 16 at point (n 1 , t 1 ) = (0, 0) with the shift n 0 = 1 3 , which is an on-site RW (see Fig. 1a). The minima amplitude attains 0 at two sites (n 2 , respectively. Moreover, we find that the lower peak amplitude of the first-order RW can reach at two adjacent lattice sites when n 0 = 5 6 , which is called inter-site RW (see Fig.1b). Through detailed calculation, we find that the higher-order discrete NLS equation (2) has the identical amplitude but different center points with the same background wave plane comparing with the fundamental RW solution in the AL equation [17]. Next, we consider the effect of higher-order term γ and spectrum parameter λ on RWs. Figure 2a, b, c show that the first-order RWs become narrower with the increase of nonlinear term parameter γ but the peak does not changed. When γ → ∞, the first-order RWs can concentrate the energy. On the conversely, when γ → 0, Eq. (2) reduces to the AL equation, and the RWs approach the fundamental RWs of the AL equation. Altering the parameter λ, we see that the amplitudes of the fist-order RWs increase with the spectrum λ increase (see Fig. 2d, e).
The numerical simulation with random noise is an effective method to test the stability of the system [30,31]. In what follows, we study the dynamical behaviors of the first-order RW solutions by numerical simulation with the initial conditions and perturbation for Eq. (2). Figure 3a is the exact first-order RW solution (15). Figure 3b, c are the profiles of the numerical simulation, which exhibit the time evolution of the RWs with initial condition and the perturbation of the initial solution with 2% amplitude as random noise at t ∈ (−1, 1), respectively. The corresponding results show that the numerical simulations of the first-order RW solution can well agree with the exact RW solution (15) besides a little weak oscillation near the edges with the perturbation case.
• For the case d 1 = f 1 = 0, the strong interaction happens that the RWs have four minimum points and five local maximum including a biggest peak at the center of the wave packets (see Fig. 4a). • For the case d 1 = 0 or f 1 = 0, the second-order RWs split into three first-order RWs, whose centers become a rotating triangle, and the whole profiles have three local maximum and six minimum points (see Fig. 4b, c).
Moreover, adjusting the parameters freely, we find that the area of the triangle increases with the increase of the parameters |d 1 | or | f 1 | and | f 1 | can control the rotation of the triangle RWs. Next, we give the dynamical property for the secondorder RWs by the numerical simulation. Figure 5a, d are exact second-order RW solutions with different parameters d 1 and e 1 . Figure 5b, c show that the numerical simulation of the strong interaction (i.e.,d 1 = 0, f 1 = 0) can well agree with the exact solution except for weak oscillations at t > 0.4 (see Fig. 5c). For the weak interaction case (i.e.,d 1 = 10, f 1 = 0), we find that the wave propagation can also match the exact solution well. However, if we add the random noise (2%) to the initial solution, the weak interaction displays serious oscillations after time exceeds 0.2, which may be due to the main energy distribution [17].
Case 3: third-order RW solutions When N = 3, by the formula (13) and take the special spectral parameters λ = 7 4 with c = 33 56 , then the thirdorder discrete RW solution is obtained as where Fig. 5 The second-order RW solutions (17). Exact solutions with a d 1 = f 1 = 0 and d d 1 = 10, = f 1 = 0. b and e the numerical simulation using exact solutions (17) at t = −1 as initial conditions. d and f numerical simulations by adding random noise with amplitude 2% to the exact solutions (17) as initial conditions and [3] 1 and [3] 2 change to [3] but the first row and second row in the [3] are replaced by (φ 1 [4,0], , respectively. The exact expression of three-order RW solution is so clumsy that we omit it here. We just give its structural analysis corresponding to the four different parameters (d 1,2 , f 1,2 ).
• For the case d 1 = 10, d 2 = f 1,2 = 0, the thirdorder RWs split into six first-order RWs, which form a triangular pattern (see Fig. 6b); • For the case d 2 = 10, d 1 = f 1,2 = 0, the thirdorder RWs also split into six first-order RWs, which array to a rotating pentagon pattern with a first-order RWs located at the center (see Fig. 6c). Now we study the dynamical behaviors for the third-order RWs (18) by the numerical simulation. Here, we only consider the strong-interaction case (see Fig. 6a) and weak interaction case (see Fig. 6b). Figure  7a, b show that the strong-interactions of third-order RWs almost agree with the exact solution (18). If a small noise adds to the exact solution (18) in stronginteraction case, the wave propagation behaves well except a small bulge around the edges (see Fig. 7c) but the amplitude is obviously lower than the one's of the exact solution and numerical simulation case (see Fig. 7a, b). On the other hand, no matter what we add a noise or not to the initial condition, the wave propagations of weak interaction of third-order RWs display strong oscillations (see Fig. 7d-f). We infer that the dispersed energy of the third-order RWs can more easily lead to the disorder than the strong interaction case.
Assume that real equation (21) exists the following complex solution (q 1,n (t), q 2,n (t)) = (q (0) 1,n , q (0) 2,n )e gt+ikn , where g is the MI gain k is an arbitrary real wavenumber and q  We point out here the MI takes place when expression (24) is positive. The MI condition g 2 > 0 holds as cos k > 1 − c 2 1 + c 2 , , and k = 2mπ, m ∈ Z . Figure 8 shows that the growth rate g(k) becomes larger and larger as the amplitude c increases, meanwhile, for the fixed amplitude c = 2, g max (k) also gets larger with the increase of parameter γ .

Conclusions
In this paper, we have studied a higher-order integrable discrete NLS equation by the generalized discrete (1, N − 1)-fold Darboux transformation. The discrete higher-order RW solutions are given by determinants. We have analytically studied the dynamical behaviors of discrete RW solutions, which exhibits abundant patterns including on-site, inter-site, the rotating triangle and pentagon structures. Comparing with the discrete NLS equation, the first-order RWs of integrable higher-order discrete NLS equation have the identical peaks but different center points with the same plane-wave amplitude. We also find that the nonlinear term parameter γ can control energy density of the RWs. By means of numerical simulation with the time evolutions of the RW solutions, we reveal that the strong interaction RWs are more stable on the wave propagation than the weak interaction case. Finally, the modulation instability condition of the background wave solutions are given.