Nth-order smooth positon and breather-positon solutions for the generalized integrable discrete nonlinear Schrödinger equation

In this paper, we investigate the smooth positon and breather-positon solutions of the generalized integrable discrete nonlinear Schrödinger (NLS) equation by the degenerate Darboux transformation (DT). Starting from the zero seed solution, the Nth-order smooth positon solutions are obtained by degenerate DT. The breather solutions including Akhmediev breather, Kuznetsov-Ma breather and space-time periodic breather are derived from the nonzero seed solution. Then the breather-positon solutions are constructed by gradual Taylor series expansion of the eigenfunctions in breather solutions. We study the effect of the coefficient of nonlinear term on these discrete smooth positon solutions and breather-positon solutions, which demonstrates that the interacting region of soliton-positon and breather-positon are highly compressed by higher-order nonlinear effects, but the distance between the two positons has an opposite effect in two waveforms.

The positon solution first proposed by Matveev for the Korteweg-de Vries (KdV) equation is a slowly damped oscillation solution [26], which has the special property of being superreflectionless [27]. The essential feature of the positon is that it is generated by a positive spectral singularity embedded in the continuous spectrum [28]. Meanwhile, a large number of singular positon solutions are found in nonlinear integrable system, such as the Sine-Gordon equation [29], the defocusing modified KdV (mKdV) equation [30,31], the Toda lattice [32] and the Hirota-Satsuma coupled KdV equation [33]. The smooth positon solution has been constructed for the soliton equations of the derivative NLS equation [34], the second-type derivative NLS equation [35], the focusing mKdV equation [36], the complex mKdV equation [37] and the higher-order Chen-Lee-Liu equation [38]. Subsequently, the degenerate solutions of the spatial discrete Hirota equation have been reported in [20].
(1) is the following higher-order NLS equation [40] iq t + q x x + 2|q| 2 q + γ q x x x x + 8|q| 2 q x x + 6q * q 2 which can describe the energy transfer in higher-order excitation of α-helical proteins [39,41]. A great number of exact solutions including the multi-soliton solutions, multi-positon solutions and higher-order rogue wave solutions have been investigated in [42][43][44]. However, to our knowledge, the Nth-order smooth positon solutions and Nth-order breather-positon solutions of this generalized integrable discrete NLS equation have not been reported. Therefore, it will be an interesting problem to investigate the smooth positon solutions, breather-positon solutions and the affect of the coefficient of higher-order nonlinear terms on these solutions. This paper is organized as follows. In Sect. 2, we present the smooth positon solutions of Eq. (1) by the degenerate DT and then investigate the effect of the coefficient of nonlinear term on these solutions. In Sect. 3, the discrete breather solution of Eq. (1) is derived from the nonzero seed solution. In Sect. 4, the breather-positon solution of Eq. (1) is constructed from the degenerated breather solution. The conclusion is given in the last section.

Positons of generalized integrable discrete NLS Equation (1)
Referring to [19], the Lax pair of the generalized integrable discrete NLS Equation (1) reads where the symbol E is a translation operator defined as E f (n) = f (n + 1). The matrices L n and N n have the following forms where The zero curvature condition L n,t = N n+1 L n − L n N n of the linear spectral problem (3) yields the generalized discrete NLS equation (1).
The determinant representation of the N-fold Darboux transformation for Eq.(1) has been given by [19] q n [N ] = −q n T where with .

The expressions of [N ]
n,1 and [N ] n,2 have the same structure with [N ] n except the replacement of the first column and the second column, respectively, in [ n,2 ) T . Starting from the zero seed solutions q n = 0, the linear system (3) becomes where Solving the linear spectral problem (7), we have where Y (λ) = n ln λ + ω(λ)t, Z (λ) = n ln λ −1 − ω(λ −1 )t and c j ( j = 1, 2) are arbitrary complex parameters. When N = 1, taking the eigenfunctions ψ n, j ( j = 1, 2) as follows: and inserting them into (5), we get one soliton solution where W R , W I are the real and imaginary parts of where In following, we present the derivation of the positon solutions of the generalized integrable discrete NLS equation. By setting the degenerate limit [20] λ j → λ 1 + ( j = 2, 3, 4, ..., N ), we perform the higherorder Taylor series of the eigenfunctions (8) around = 0. Starting from the zero seed solution q n = 0, the N-order soliton-positon solutions of the generalized integrable discrete NLS Eq.(1) can be written as where When N = 1, the degenerate limit of the determinant representation is trivial, which leads to the one soliton solution (10). To derive the second-order smooth soliton-positon solution, we set N = 2 in formula (12) and then we have q [2] n− p = − where The positon solution is smooth traveling wave solution, which is expressed as a mixed form of exponential function and polynomial. We show the second-order smooth positon solutions in Figs. 1. To study the effect of the coefficient of nonlinear term γ in the framework of the above discrete positon solutions, we first plot the solution (17) with γ = 0, which becomes the secondorder positon solution of the Ablowitz-Ladik equation (see Fig. 1a). Increasing the value of the parameter γ , we observe that the distance between two positons gets large and the interactive region becomes tight (see Fig.  1b and c). Thus, we conclude that the central region of the second-order positon interaction is compressed with the increase of the parameter value γ while the distance of the positons goes far from each other, which is opposite to the one of the generalized NLS equations [42]. In addition, the trajectories of the positon solution also change.
The third-order positon solution of the generalized integrable discrete NLS equation (1) can be obtained from the solution (17) with N = 3. Since the exact solution of the third-order positon is very complicated, we do not write it here. The third-order positon solution is plotted in Fig. 2. We analyze the dynamical behavior of the third-order positon solution with the effect of the coefficient of nonlinear term γ . For γ = 0, it degenerates into the third-order positon solution of the Ablowitz-Ladik equation (see Fig. 2a). We increase the parameter value γ to 0.4. Figure 2b and c show that the interaction regions are compressed and the directions are in growing incline. The distance between the three Fig. 1 The evolution of second-order smooth positon solution |q [2] n− p | with λ 1 = 1.2 + 0.25i,c

Breather solution
To determine the breather solution of the generalized integrable discrete NLS equation (1), we solve the linear spectral problem (3) with nonzero seed solution, that is q n = ce ikt , where k = 2c 2 (3c 2 γ + h 2 )/ h 4 and c is a real constant. Then the eigenfunction expressions can be written as where By denoting η 1 = P 1 n + 1 t, where P 1 = ln ν + 1 (λ 1 ) ν − Fig. 2 The evolution of third-order smooth positon solution |q [3] n− p | with λ 1 = 0.35 are the contour plots of the upper levels q [1] n−b = e ikt G 1 F 1 where We point out that the three situations of P 1R = 0, P 1R = 0 and 1R = 0, 1R = 0 and P 1R = 0 give rise to the spatially periodic breather, time-periodic breather and space-time periodic breather, respectively.
In particular, setting λ 1 = e τ then eigenvalues ν ± 1 satisfy the following quadratic equation In particular, choosing c = 1, γ = 0.2 and λ 1 = 0.5, which yields P 1R = 0, we obtain the exact Akhmediev breather solution (see Fig. 3a). In order to obtain the time-periodic breather, we shall have respect for the corresponding relationships P 1R = 0 and 1R = 0. If we take the parameters c = 1, γ = 0.3, λ 1 = 0.34i, then we have P 1R ≈ 1.65 and μ + 1R = μ − 1R . We depict the solution (15) in Fig. 3b. The plot confirms that the solution is periodic in n and localized in t, which is called the Kuznetsov-Ma breather. If choosing c = 1, γ = 0.5, λ 1 = 1.8+0.23i, which yields P 1R ≈ −0.2 and 1R ≈ −5.63, the space-time periodic breather is derived (see Fig. 3c). The formula (5) with N = 2 produces the second-order breather solution of Eq. (1) q [2] n−b = ce ikt where [2] n,1 , and [2] n,1 and [2] n,2 are described by [2] n in which the first column and second column are changed into n,2 ) T , separately. The profiles of the two breather solutions (17) are displayed in Fig. 4, which exhibits the dynamical interaction of two Akhmediev breather, Akhmediev breather and Kuznetsov-Ma breather and two space-time periodic breather, respectively. We remark here that each component of the second-order breather obeys the periodic constraint rules as the first-order breather on spectral parameter. Substituting the eigenfunction (14) into the N-fold DT (5) and taking the limit λ i → λ 1 + (i = 2, 3, · · · , N ), we derive the N-order breather-positon solution of Eq. (1) where As we have known, the breather-positon solution is a new state of breather solution by the degenerate limit and the higher-order Taylor expansion of the eigenfunctions. Thus, the higher-order breather-positon solution can be regarded as the intermediate of the higher-order rogue wave. Moreover, each component of the higherorder breather-positon solution has the same period but different phases due to the unique limit process. When N = 2, the formula (18) gives the secondorder breather-positon solution, but we do not write the explicit expressions as it has the tedious form of exponential function and polynomial. Figure 5 describes the dynamic evolution of space-periodic breather-positon, time-periodic breather-positon and space-time periodic Next, we consider the effect of the coefficient of nonlinear term γ on the second-order breather-positon solution. When γ = 0, Fig. 6a reduces to the secondorder breather-positon of the Ablowitz-Ladik equation. When we gradually increase the value of the parameter γ to 0.4, the distance between the two breatherpositon and the central region is tightly compressed, which is different from the case of the two solitonpositon. Meanwhile, when we increase the parameter γ , the period of the breather-positon solution remains unchanged due to the fact that the wave number keeps constant (see Fig. 6b and c).

Conclusion
In this paper, we construct the degenerate solutions of the generalized integrable discrete NLS equation by degenerate DT. The smooth positon solution has been derived by the higher-order Taylor expansion with the degenerate eigenvalue limit of the N-soliton solution. For the soliton-positon solution, we have shown that the central region of interaction is compressed by the coefficient of nonlinear term γ but the distance of the higher-order positon increases with the increase of γ . From the nonzero seed solution, we obtain the breather solutions of the generalized integrable discrete NLS equation including the space-periodic breather, timeperiodic breather and space-time periodic breather. Then the breather-positon solutions of the generalized integrable discrete NLS equation are derived from the breather solutions. The coefficient of nonlinear term γ exhibits the highly compressed both the interaction region and the distance between the two breatherpositon solution, while the period of the breatherpositon is unaffected. It is hoped that this work may Fig. 6 The evolution of second-order breather-positon solution |q [2] n−bp | with λ 1 = 2, c Data availability All data generated or analyzed during this study are including in this published article.