Various Soliton Solutions and Asymptotic State Analysis for the Discrete Modified Korteweg-de Vries Equation

Under investigation is the discrete modified Korteweg-de Vries (mKdV) equation, which is an integrable discretization of the continuous mKdV equation that can describe some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Through constructing the discrete generalized ðm,N −mÞ-fold Darboux transformation for this discrete system, the various discrete soliton solutions such as the usual soliton, rational soliton, and their mixed soliton solutions are derived. The elastic interaction phenomena and physical characteristics are discussed and illustrated graphically. The limit states of diverse soliton solutions are analyzed via the asymptotic analysis technique. Numerical simulations are used to display the dynamical behaviors of some soliton solutions. The results given in this paper might be helpful for better understanding the physical phenomena in plasma and nonlinear optics.


Introduction
Nonlinear partial differential equations (NPDEs) such as Burgers equation, KdV equation, mKdV equation, and nonlinear Schrödinger (NLS) equation can describe many important physical phenomena in nonlinear optics, acoustic in the nonharmonized lattice, deep water waves, plasma environments, and so on (see [1][2][3][4][5] and the references therein). The explicit exact solutions, especially soliton solutions, of the NPDEs play a vital role in many practical applications [6]. The soliton structures are formed when an exact balance between nonlinearity and dispersion effects in NPDEs takes place, and the KdV-type equation can usually describe the evolution of the unmodulated soliton in the small amplitude [6]. Another type of envelope soliton (dark or bright soliton and rogue wave) is formed when wave group dispersion is in complete balance with the nonlinearity of the medium, and this type of envelope soliton is a localized modulated wave packet whose dynamics are governed by the family of the NLS equation [6]. The Gardner equation is also called combined KdV-mKdV equation which is widely applied in various branches of physics [7]. As is well known, the KdV equation possesses bright soliton structures; while the mKdV equation can admit bright solitons and shock wave solution, compared with the KdV and mKdV equations, the Gardner equation admits solitons of the hyperbolic functions and kink solutions [7]. The soliton collision is an interesting and important nonlinear phenomenon in the nonlinear medium, nonlinear dynamics of soliton collisions, and soliton phase shifts in the KdV, and mKdV equations are taken for discussion [6,8,9]. However, to the authors' knowledge, for the continuous KdV-type equation, this soliton collision phenomenon and phase shift analysis have been discussed more, but for the discrete KdVtype equation, the relevant research for soliton collision and phase shift is poor, so it is a meaningful research topic to extend this collision phenomenon to discrete nonlinear lattice equations.
Recently, discrete integrable nonlinear lattice equations, as spatially discrete analogues of NPDEs, have drawn widespread attention due to their appearance in a variety of fields such as the propagation of optical pulses in nonlinear optics, Langmuir wave in plasma physics, nonlinear lattice dynamics, population dynamics, anharmonic lattice dynamics, Bose-Einstein condensates, and propagation of electrical signals in circuits [10][11][12][13][14]. Searching for explicit exact solutions, especially soliton solutions, is used for depicting and explaining such nonlinear phenomena described by the discrete nonlinear lattice models. Methods of constructing the soliton solutions of the discrete integrable models have been proposed and developed such as the discrete inverse scattering method [13], the discrete Hirota bilinear formalism method [14,15], and the discrete Darboux transformation (DT) method [16][17][18][19][20][21][22][23][24][25]. Among them, the discrete DT based on corresponding Lax representation is a useful technique to solve the discrete nonlinear models and its main idea is to keep the corresponding Lax pair of these discrete equations unchanged. Recently, a discrete generalized ðm, N − mÞ-fold DT method has been proposed by one of the present authors of this paper [24,25]. Compared with the common discrete DT which can only give the usual soliton (US) solutions, the discrete generalized ðm, N − mÞ-fold DT [24,25] is a generalization of the common discrete DT and the main advantage of this method is that it can give not only the US solution but also the rational solutions (e.g., rouge wave solutions and rational soliton (RS) solutions) and mixed interaction solutions of US and rational solution [24,25]. For the US solutions of discrete nonlinear models, there have been a lot of literature studies [16][17][18][19][20][21][22][23] but the study of rational solutions and mixed interaction solutions of US and rational solutions is still not sufficient and systematic. As far as we know, the asymptotic behaviors of RS solutions and mixed interaction soliton solutions of US and RS by using asymptotic analysis have not been studied yet. Therefore, in this paper, via the discrete generalized ðm, N − mÞ-fold DT and asymptotic analysis techniques, we are going to focus on the following discrete mKdV equation [26]: where q n = qðn, tÞ stand for the real function of the discrete variable n and time variable t, and q n,t = dq n /dt. Equation (1), which possesses more nonlinear terms, is different from the discrete mKdV equation in [13,23,[27][28][29][30]. Moreover in [26], Suris has given an Ablowitz-Ladik hierarchy including equation (1), whose Lax pair admits with M 11 = λ 4 + q n q n−1 − 2 ð Þ λ 2 + q 2 n + q n−2 q n À Á q 2 n−1 + q 2 n q n+1 − 2q n + q n+1 À Á q n−1 + q n−2 q n − q n−1 q n λ 2 , where φ n = ðϕ n , ψ n Þ T is a basic solution of equation (2) (the superscript T means the vector transpose), λ is the spectral parameter, and the shift symbol E is defined by Ef n = f n+1 . It is easy to find that the zero curvature equation can be calculated by compatible condition φ n,t = φ t,n , which is subject to equation (1). If we assume that q n = εqðnεÞ in equation (1) and to rescale time t ↦ t/2ε 2 , then, to send ε ⟶ 0, thus, equation (1) can be reduced to the following continuous mKdV equation [26,31,32]: The above process is called the continuous limit from equation (5) to equation (1). The mKdV equation (5) is a model to describe acoustic in the nonharmonized lattice and also can be used to study solitary waves in dusty plasmas and fluctuations in nonlinear optics etc. [3][4][5][33][34][35][36][37][38][39], so it is important to study equation (5) in the physical background and practical significance. In [13,40,41], the authors point out that the discrete models can preserve some properties of its corresponding continuous equations, so equation (1), taken as the corresponding discrete equation of equation (5), may also have potential applications in describing some physical phenomena such as dust solitary waves in dust plasma and fluctuations in nonlinear optics, which suggests that the study of equation (1) also might have important theoretical significance and practical application value. Based on what we know, the US solutions, RS solutions, their mixed interaction soliton solutions via the discrete generalized ðm, N − mÞ-fold DT, and associated soliton limit states via the asymptotic analysis technique for equation (1) have not been investigated before. Therefore, in this paper, we will further investigate equation (1) via the discrete generalized ðm, N − mÞ-fold DT and we will study the asymptotic behaviors at infinity of diverse soliton solutions via the asymptotic analysis technique, via the discrete generalized ðm, N − mÞ-fold DT. The rest of this paper is organized as follows. In Section 2, based on the known Lax representation (2), the discrete generalized ðm, N − mÞ-fold DT of equation (1) will be constructed and 2 Advances in Mathematical Physics formulated. In Section 3, the different special cases of the discrete generalized ðm, N − mÞ-fold DT method are used to derive various soliton solutions such as US solutions, RS solutions, and their mixed interaction soliton solutions and we will analyze the limit states of such obtained soliton solutions via asymptotic analysis. Meanwhile, we also will use numerical simulations to watch the solutions' dynamical behaviors so that we can comprehend or even predict the physical properties of solutions more clearly. Some conclusions are given in the final section.

Discrete Generalized ðm, N − mÞ-Fold DT
In this section, we will build up the discrete generalized ðm , N − mÞ-fold DT of equation (1). With Lax pair (2), we consider a gauge transformation in the following form: which requires thatφ n = ð e ϕ n ,ψ n Þ T must be subject toφ n = L nφn ,φ n,t =M nφn according to the knowledge of DT, wherẽ L n ,M n have the same forms as L n , M n except by replacing its q n withq n . Moreover,L n ,M n and L n , M n also admit the following identities: To this aim, we construct a special N-order Darboux matrix T n defined by in which N is a positive integer and a ð2jÞ n and b ð2j+1Þ n are some unknown functions of n, t and can be determined by where N = n + ∑ n j=0 m i ði = 1, 2, ⋯, nÞ and T ðjÞ n is given by By explaining the above research processes and conditions, we conclude the following generalized ðm, N − mÞ-fold DT theorem of equation (1): Theorem 1. Let φ n ðλ i Þ = ðϕ n ðλ i Þ, ψ n ðλ i ÞÞ T be m column vector solutions of Lax pair (2) with the special parameter λ i ði = 1, 2, ⋯, mÞ and the initial seed solution q 0 of equation (1), and then, the generalized ðm, N − mÞ-fold DT from the old solution q n to the new oneq n is given bỹ where Δ ðiÞ m i +1 = ðΔ ðiÞ j,s Þ 2ðm i +1Þ×2N , in which Δ ðiÞ j,s ð1 ≤ j ≤ 2m i + 2, 1 ≤ s ≤ 2NÞ be given by following formulae: 3 Advances in Mathematical Physics and ΔB ðN−1Þ are given by the determinant Δ ε N replacing its ðN + 1Þth columns by the column vector ðg ð1Þ ,⋯,g ðnÞ Þ T with g ðiÞ Remark 2. It is worth pointing out that m in the notation "ðm, N − mÞ-fold" means that the number of the distinct spectral parameters and N − m means the sum of the orders of the Taylor series of the vector eigenfunction φ n . If n = N and m i = 0 ð1 ≤ i ≤ NÞ, the generalized ðm, N − mÞ-fold DT is converted to the ðN, 0Þ-fold DT which includes the usual N-fold DT, from which we can develop the usual N-soliton solutions starting from zero seed or nonzero constant seed solution of equation (1). If m = 1, the above generalized ðm, N − mÞ-fold DT reduces to the generalized ð1, N − 1Þ-fold one, from which we can derive higher-order RS solutions of equation (1). If m = 2, the above generalized ðm, N − mÞ-fold DT reduces to the generalized ð2, N − 2Þ-fold one, from which we can derive mixed soliton solutions of the US and RS to equation (1). It is not hard to find that the N-fold DT, generalized ð1, N − 1Þ -fold DT, and generalized ð2, N − 2Þ-fold DT are three special cases of the generalized ðm, N − mÞ-fold DT. Besides, if 2 < m < N, one can use the generalized ðm, N − mÞ-fold DT to give complex mixed interaction soliton solutions. For more details of Theorem 1, the reader can refer to [23][24][25] and the references therein, so we omit the detailed proof here.

Applications of the Generalized ðm, N − mÞ-Fold DT
In the previous section, we have constructed the discrete generalized ðm, N − mÞ-fold DT of equation (1). So, in this section, we will use the generalized ðm, N − mÞ-fold DT to get various soliton solutions, like US solutions, RS solutions, and their mixed soliton solutions and then analyze the elastic interaction and limit state of such solitons by using asymptotic analysis.

Usual N-Soliton Solutions and Their Asymptotic
Analysis. We can get usual soliton solutions from the vanishing background by using N-fold DT, namely, in the condition of m = N in discrete generalized ðm, N − mÞ-fold DT.
Substituting the initial solutions q n = 0 into Lax pair (2) can give its basic solution: From equation (10), we can derive the exact soliton solutions of equation (1). In order to comprehend them more intuitively, the evolution structures of soliton solutions are shown in Figures 1-4 when N=1,2,3.
(I) when N=1, let λ = λ 1 ; based on the N-fold DT, we can get the onefold exact solution as where b ð1Þ n = Δb ð1Þ n /Δ n , in which n+1 is obtained from b ð1Þ n by replacing n with n + 1. After bringing (14) into equation (15), we will get the usual one-soliton solution. In order to explore its physical characteristics, this solution can be rewritten as where Þt and λ 1 > 0 is a spectrum parameter. In this special one-solution form, we can easily explore its important physical quantities such as amplitude, width, velocity, wave number, primary phase, and energy, which are listed in Table 1. Here, the energy of q n is defined by E q n = Ð ∞ −∞ q 2 n dn. From the above analysis, we can find that q n is the bellshaped bright one-soliton when 0 < λ 1 < 1 and anti-bellshaped one-soliton when λ 1 > 1. When λ 1 = 1/2 and λ 1 = 2, Figure 1 presents the evolution structures of bell-shaped one-soliton and anti-bell-shaped one-soliton, respectively.
Next, adopting the finite difference method [42], we can simulate the soliton solutions numerically, which can show the dynamical behaviors and propagation stability of onesoliton solutions more clearly. Figure 2 shows the exact one-soliton solution (17) of equation (1), time evolutions using exact one-soliton solution (17), and the results of adding 2% and 5% perturbations to the exact one-soliton solution. As can be seen, Figures 2(a) and 2(b) show that the representation of time evolutions for solution (17) without noise almost keeps in line with the exact solution (17) in time t ∈ ð−0:5,0:5Þ, which shows the accuracy of our numerical simulation. Compared with the unperturbed solution, Figures 2(c) and 2(d) exhibit that if we add 2% and 5% perturbations to the initial exact solution, the wave propagation performs a relatively small oscillation in time t ∈ ð−0:5,0:5Þ, that is to say that the numerical results in Figure 2 show that      (II) when N=2, let λ = λ i ði = 1, 2Þ; we can get the twofold exact solution from the transformation (10) as where b ð1Þ n = Δb ð1Þ n /Δ n , in which while b ð1Þ n+1 is obtained from b ð1Þ n by replacing n with n + 1. For the sake of analysis, solution (18) can be rewritten as According to the ideas of [18,[43][44][45], we perform asymptotic analysis of solution (20), from which we can easily work out the limit state expressions of solutionq n before and after the interaction. And the detailed physical quantities are listed in Table 2.
Before the interactions ðt ⟶ −∞Þ, where q − 1 , q − 2 are the asymptotic state expressions ofq n before the interaction.
After the interactions ðt ⟶ +∞Þ, where q + 1 , q + 2 are the asymptotic state expressions ofq n after the interaction.
From the above analysis, we can see that the characteristics and embodiment of the two-soliton solution in the physical field can be summarized as follows: (i) the amplitude, velocities, and energy of q n remain unchanged before and after the interactions; (ii) after the soliton interactions, the phase shifts of two solitons are −2 ln jðλ 2 1 − λ 2 2 Þ/ðλ 2 1 λ 2 2 − 1Þj and 2 ln jðλ 2 1 − λ 2 2 Þ/ðλ 2 1 λ 2 2 − 1Þj; and (iii) the soliton width, wave number, and wave shapes are related to two spectrum parameters λ i ði = 1, 2Þ, and the relationship between shapes and parameters are shown in Table 3.
The corresponding evolution plots of solution (18) or (20) can be also elaborated clearly. When the parameters λ 1 = 2 and λ 2 = 1/3, we can see that the overtaking elastic interactions between two unidirectional anti-bell-shaped solitons on the vanishing background in Figures 3(a1)
10 Advances in Mathematical Physics

and 3(c2) is obviously higher than that in Figures 3(a1)-3(b2). In Figures 3(c1) and 3(c2)
, we can clearly see that the soliton interaction occurs at t = 0 and the high amplitude occurs near the origin; if we choose n = t = 0, then, from solution (20), we haveq n = ð1 − ð1/λ 1 λ 2 ÞÞðλ 1 + λ 2 Þ. As λ 1 λ 2 approaches 1, the value ofq n is small. However, when both λ 1 and λ 2 are greater than 1, the value ofq n is larger. Therefore, we can infer that the choice of parameters λ 1 and λ 2 determines the amplitude, shape, and energy of the soliton.
(III) when N=3, let λ = λ i ði = 1, 2, 3Þ; we can get the threefold exact solution from the transformation (10):q where b ð1Þ n = Δb ð1Þ n /Δ n , in which while b ð1Þ n+1 is obtained from b ð1Þ n by replacing n with n + 1. Because of the complexity of the results, we do not discuss the asymptotic analysis of three solitons anymore. But we will explore the shapes of three-soliton solutions in different parameters. When the parameters λ 1 = 5/2, λ 2 = 1/2, and λ 3 = 1/2, we can see the overtaking elastic interactions among unidirectional bell-shaped three solitons on the vanishing background in Figures 4(a1) and 4(a2); when we choose the parameters λ 1 = 2, λ 2 = 2/5, and λ 3 = 3, the overtaking elastic interactions among unidirectional anti-bell-shaped three solitons on the vanishing background d are shown in Figures 4(b1) and 4(b2); when we choose the parameters λ 1 = 5/2, λ 2 = 2, and λ 3 = 1/3, the overtaking elastic interactions among unidirectional two bell-shaped solitons and one anti-bell-shaped soliton on the vanishing background are shown in Figures 4(c1) and 4(c2); when we choose the parameters λ 1 = 2, λ 2 = 2/ 5, and λ 3 = 1/3, the overtaking elastic interactions among two unidirectional anti-bell-shaped solitons and one bellshaped soliton on the vanishing background are shown in Figures 4(d1) and 4(d2). Using the same analysis method as the high amplitude of the two-soliton solution, we can also analyze the reason for the high amplitude of the soliton in Figures 4(d1) and 4(d2) by choosing three spectral parameters λ 1 , λ 2 , and λ 3 , so we will not do a detailed analysis here.

Discrete RS Solutions and Their Asymptotic Analysis.
In this section, we will use the discrete generalized ð1, N − 1Þ -fold DT to construct RS solutions with the initial nonzero constant seed solution q n = c. Because of the change of the initial background, we must rewrite solution (14) as where τ ± = 1 2λ 11 Advances in Mathematical Physics and C 1 , C 2 are arbitrary constants. Next, we fix the spectral parameter λ = λ 1 + ε 2 with λ 1 = c + ffiffiffiffiffiffiffiffiffiffiffi c 2 + 1 p . In particular, if we take C 1 = −C 2 = 1/ε, c = 3/4 and expand the vector function φ in equation (25) as two Taylor series at ε = 0, let ξ = 64n + 675t, and then, we can obtain and the rest φ ðjÞ n ðj ≥ 3Þ are omitted here.
(I) when N=1, we can give the first-order RS solution of equation (1) by using the discrete generalized ð1, 0Þ -fold DT asq where a ð0Þ n = Δa ð0Þ n /Δ n and b ð1Þ n = Δb ð1Þ n /Δ n , in which from which we can conclude or infer the following physical characteristics of RS solution (32): (i)q n ⟶ 3/4 as n ⟶ ±∞ or t ⟶ ±∞, which clearly shows that the RS solution turned out to be what it was at the beginning of initial background 3/4. It is also worth to notice thatq n reaches the minimum of −63/16 when the RS solution along the line 3ξ − 64 = 0, namely, the amplitude of this solution is 63 /16 (ii) The widths, velocities, wave number, and primary phases of this solution are 2/3, 675/64, 3/2 and −1/ 2, respectively (iii) After removing the background 3/4 of solution (32), similar to the analysis of one-soliton solution (17), we can also calculate the energy 1875/256π of solution (32) What is more, we plot its structure figures as shown in Figures 5(a1) and 5(a2).
(II) when N=2, we can give the second-order RS solution of equation (1) via the generalized ð1, 1Þ-fold DT asq 12 Advances in Mathematical Physics For the sake of discussion, the simplification form of solution (33) is listed as follows: where In order to understand the physical properties of the second-order RS solution of equation (1) better, we can still perform the asymptotic analysis. For convenience, let ζ = 64n + 675t − 64ð1275/16Þ 1/3 t 1/3 ; we can calculate the limits ofq n in (35), which only gives the following one limiting state when t ⟶ ±∞: From the above calculation, we can see that q ± n is the result ofq n when t approaches infinity and it is worth noticing that the result of positive infinity is the same as that of negative infinity. Meanwhile, the asymptotic expression (37) clearly shows that the dark RS solution q ± n reaches the minimum of −63/16 along the curve 3ζ − 64 = 0. The structures of second-order RS solution are shown in Figures 5(b1) and 5(b2), from which we can see that they are consistent with the above analysis results.
(III) when N=3, we can give the discrete third-order RS solution of equation (1) via the generalized ð1, 2Þ -fold DT asq The simplification form of solution (38) can be also calculated by using symbolic computation. But it is so complicated that therefore, we omit it here. Besides, the asymptotic analysis can be still performed. Similar to the above analysis process of second-order RS, let ζ 1 (ii) If ζ 2 = 64n + 675t − 64ðð6375/32Þ + ðð3825 ffiffi ffi 5 p Þ/32Þ Þ 1/3 t 1/3 = Oð1Þ, from ζ 1 = ζ 2 + bt 1/3 , we can get ζ 1 ⟶ ±∞ as t ⟶ ±∞; then, calculating the limit of solutionq n in (38) gives the following asymptotic state expression as From the above calculation, we can see that q ± 1 and q ± 2 are the two results ofq n when t approaches infinity and 3 ζ 1 − 64 = 0 and 3ζ 2 − 64 = 0 are also the two center trajectories of solutionq n . Meanwhile, the form of the third-order 16 Advances in Mathematical Physics RS solution at infinity clearly shows that q ± 1 and q ± 2 reach the minimum along curves 3ζ 1 − 64 = 0 and 3ζ 2 − 64 = 0, respectively. The structure of the third-order RS solution is shown in Figures 5(c1) and 5(c2), which is consistent with the result of above analysis.
Remark 3. The point here is that when we perform asymptotic analysis to the usual two-soliton and three-soliton solutions, the final two and three solitons keep their shape before and after the collisions, and for asymptotic analysis to second-order and third-order RS solutions, we find that there is one first-order RS left at infinity for second-order RS solutions and the other first-order RS disappears at infinity, while there are two first-order RSs left at infinity for third-order RS solutions and the third RS disappears. For higher-order RS solutions, there is always a RS solution that disappears at infinity, which is a little like rogue wave solutions that are asymptotic to the background wave at infinity, and we find that these new phenomena about RS are inter-esting and quite different from those of the USs, which is worthy of further investigation.
Similar to numerical simulation of the usual one-soliton solution in the previous subsection, we can also add the perturbations to RS solutions in order to show its dynamical behavior and propagation stability. Figure 6 shows firstorder RS solution (32) of equation (1), time evolutions of using first-order RS solution (32), and time evolutions of adding 0.01% and 0.05% perturbations to first-order RS solution (32). As can be seen, Figures 6(a) and 6(b) show that the representation of time evolutions for solution (32) without noise is nearly close to first-order RS solution (32) in time t ∈ ð−0:5,0:5Þ, which shows the accuracy of our numerical scheme. However, as time goes on, even without noise, there is a large fluctuation. Different from the usual one-soliton solution, Figures 6(c) and 6(d) exhibit that the evolutions of wave propagation perform the relatively large oscillations with the increase of time t if we add 0.01% and 0.05% perturbations to the initial solutions. We find that

Conclusions
In this paper, we have researched the discrete mKdV equation (1), which may be used for understanding some physical phenomena such as dynamics of anharmonic lattices, solitary waves in dusty plasmas, and fluctuations in nonlinear optics. Various kinds of soliton solutions, like US, RS, and their mixed interaction soliton solutions, have been analytically investigated and discussed, which might help to understand the abovementioned physics phenomena. We sum up the main achievement of this paper as follows: firstly, we have established the generalized ðm, N − mÞ-fold DT of equation (1) Tables 1-5. We hope that these results given in this paper will be helpful for understanding such physical phenomena in nonlinear optics, anharmonic lattice dynamics, and plasma environments.

Data Availability
The datasets used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.