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

Authors: Zhe Lin; Xiao-Yong Wen; Meng-Li Qin
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

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 generalizedm,N−m-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.


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## Body

## 1. 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 [1–5] 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 KdV-type 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 [10–14]. 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 [16–25]. 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 m,N−m-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 m,N−m-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 [16–23] 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 generalizedm,N−m-fold DT and asymptotic analysis techniques, we are going to focus on the following discrete mKdV equation [26]:
(1)qn,t=1+qn2qn+2−2qn+1+2qn−1−qn−2+qn+12qn+2+qn−qn−12qn+qn−2,where qn=qn,t stand for the real function of the discrete variable n and time variable t, and qn,t=dqn/dt. Equation (1), which possesses more nonlinear terms, is different from the discrete mKdV equation in [13, 23, 27–30]. Moreover in [26], Suris has given an Ablowitz-Ladik hierarchy including equation (1), whose Lax pair admits
(2)Eφn=Lnφn,Ln=λqn−qn1λ,φn,t=Mnφn,Mn=M11M12M21M22,with
(3)M11=λ4+qnqn−1−2λ2+qn2+qn−2qnqn−12+qn2qn+1−2qn+qn+1qn−1+qn−2qn−qn−1qnλ2,M12=qnλ3+qn+1+qn2qn−1+qn2qn+1−2qnλ+qn−2+qnqn−12+qn−2qn−12−2qn−1λ+qn−1λ3,M21=−qn−1λ3+−qn−2−qnqn−12−qn−2qn−12+2qn−1λ+−qn+1−qn2qn−1−qn2qn+1+2qnλ−qnλ3,M22=−qnqn−1λ2+qn2+qnqn−2qn−12+qn2qn+1−2qn+qn+1qn−1+qnqn−2+qn−1qn−2λ2+1λ4,where φn=ϕn,ψnT 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 Efn=fn+1. It is easy to find that the zero curvature equation
(4)Ln,t=EMnLn−LnMncan be calculated by compatible condition φn,t=φt,n, which is subject to equation (1). If we assume that qn=εqnε in equation (1) and to rescale time t↦t/2ε2, then, to send ε⟶0, thus, equation (1) can be reduced to the following continuous mKdV equation [26, 31, 32]:
(5)qt=qxxx∓6qxq2.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. [3–5, 33–39], 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 m,N−m-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 m,N−m-fold DT and we will study the asymptotic behaviors at infinity of diverse soliton solutions via the asymptotic analysis technique, via the discrete generalized m,N−m-fold DT. The rest of this paper is organized as follows. In Section 2, based on the known Lax representation (2), the discrete generalized m,N−m-fold DT of equation (1) will be constructed and formulated. In Section 3, the different special cases of the discrete generalized m,N−m-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.


## 2. Discrete Generalizedm,N−m-Fold DT

In this section, we will build up the discrete generalizedm,N−m-fold DT of equation (1). With Lax pair (2), we consider a gauge transformation in the following form:
(6)φ~n=Tnλφn,which requires that φ~n=ϕ~n,ψ~nT must be subject to φ~n=L~nφ~n,φ~n,t=M~nφ~n according to the knowledge of DT, where L~n,M~n have the same forms as Ln,Mn except by replacing its qn with q~n. Moreover, L~n,M~n and Ln,Mn also admit the following identities:
(7)L~n=Tn+1LnTn−1,M~n=Tn,t+TnMnTn−1.To this aim, we construct a specialN-order Darboux matrix Tn defined by
(8)Tnλ=λ2N+∑j=0N−1an2jλ2j∑j=0N−1bn2j+1λ2j+1−∑j=0N−1bn2j+1λ2N−2j−11+∑j=0N−1an2jλ2N−2j,in which N is a positive integer and an2j and bn2j+1 are some unknown functions of n,t and can be determined by
(9)Tn0λiφn0λi=0,Tn0λiφn1λi+Tn1λiφn0λi=0,⋯⋯,∑j=0miTnjλiφnmi−jλi=0,where N=n+∑j=0nmii=1,2,⋯,n and Tnj is given by Tnλi+ε=∑k=0NTnkλiεk, while φnj is derived by φnλi+ε=φn0λi+φn1λiε+φn2λiε2+,⋯, in which φnkλi=1/k!∂k/∂λkφnλλ=λi=1/k!∂k/∂λkϕnλλ=λi,1/k!∂k/∂λkψnλλ=λiT. By explaining the above research processes and conditions, we conclude the following generalized m,N−m-fold DT theorem of equation (1):Theorem 1.
Letφnλi=ϕnλi,ψnλiT be m column vector solutions of Lax pair (2) with the special parameter λii=1,2,⋯,m and the initial seed solution q0 of equation (1), and then, the generalized m,N−m-fold DT from the old solution qn to the new one q~n is given by
(10)q~n=qnan+10+bn+11,with
(11)an+10=Δan+10Δn,bn1=Δbn+11Δn,ΔN=detΔm1+11,Δm2+12,⋯,Δmn+1mT,where Δmi+1i=Δj,si2mi+1×2N, in which Δj,si1≤j≤2mi+2,1≤s≤2N be given by following formulae:
(12)Δj,si=∑k=0j−1C2N−2skλi2N−2s−kϕij−1−k,for1≤j≤mi+1,1≤s≤N,∑k=0j−1C4N−2s+1kλi4N−2s−k+1ψij−1−k,for1≤j≤mi+1,N+1≤s≤2N,∑k=0j−N+1C2skλi2s−kψij−1−N−k,formi+2≤j≤2mi+1,1≤s≤N,−∑k=0j−N+1C2s−2N−1kλi2s−2N−k−1ϕij−N−1−k,formi+2≤j≤2mi+1,N+1≤s≤2N,and ΔBN−1 are given by the determinant ΔNε replacing its N+1th columns by the column vector g1,⋯,gnT with gi=gji2mi+1×1 in which
(13)gji=−∑k=0j−1C2Nkλi2N−kϕij−1−k,for1≤j≤mi+1,−∑k=0j−N−1C2Nkλi2N−kψij−N−1−k,formi+2≤j≤2mi+1.Remark 2.
It is worth pointing out thatm in the notation “m,N−m-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 φn. If n=N and mi=01≤i≤N, the generalized m,N−m-fold DT is converted to the N,0-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 m,N−m-fold DT reduces to the generalized 1,N−1-fold one, from which we can derive higher-order RS solutions of equation (1). If m=2, the above generalized m,N−m-fold DT reduces to the generalized 2,N−2-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 1,N−1-fold DT, and generalized 2,N−2-fold DT are three special cases of the generalized m,N−m-fold DT. Besides, if 2<m<N, one can use the generalized m,N−m-fold DT to give complex mixed interaction soliton solutions. For more details of Theorem 1, the reader can refer to [23–25] and the references therein, so we omit the detailed proof here.


## 3. Applications of the Generalizedm,N−m-Fold DT

In the previous section, we have constructed the discrete generalizedm,N−m-fold DT of equation (1). So, in this section, we will use the generalized m,N−m-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.


### 3.1. UsualN-Soliton Solutions and Their Asymptotic Analysis

We can get usual soliton solutions from the vanishing background by usingN-fold DT, namely, in the condition of m=N in discrete generalized m,N−m-fold DT. Substituting the initial solutions qn=0 into Lax pair (2) can give its basic solution:
(14)φn=ϕnψn=λneλ4−2λ2tλ−ne1λ4−2λ2t.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)
whenN=1, let λ=λ1; based on the N-fold DT, we can get the onefold exact solution as(15)q~n=bn+11,where bn1=Δbn1/Δn, in which
(16)Δn=ϕnλ1λ1ψnλ1λ12ψnλ1−λ1ϕnλ1,Δbn1=ϕnλ1−λ12ϕnλ1λ12ψnλ1−ψnλ1,while bn+11 is obtained from bn1 by replacing n with n+1.Figure 1
One-soliton solutionq~n in (15) or (17). (a1) Bell-shaped one-soliton solution with the parameter λ1=1/2. (b1) Anti-bell-shaped one-soliton solution with the parameter λ1=2. (a2, b2) The propagation processes for q~n at t=−3 (dash-dot line), t=0 (long dashed line), and t=3 (solid line).
(a)(b)Figure 2
One-soliton solution (17) with the same parameter as in Figure 1. (a) Exact solution. (b) Time evolution without a small noise. (c) Time evolution with a 2% noise. (d) Time evolution with a 5% noise.
(a)(b)(c)(d)Figure 3
Two-soliton solutionq~n in (18) or (20). (a1) Two anti-bell-shaped solitons with parameters λ1=2,λ2=1/3. (b1) Two bell-shaped solitons with parameters λ1=5/2,λ2=1/2. (c1) Bell-shaped soliton and anti-bell-shaped soliton with parameters λ1=2 and λ2=3. (a2–c2) The propagation processes for q~n at different time corresponding to (a1–c1), respectively.
(a)(b)(c)Figure 4
Three-soliton solutionq~n in (23). (a1) Bell-shaped three solitons with parameters λ1=5/2,λ2=1/2,λ2=1/3. (b1) Anti-bell-shaped three solitons with parameters λ1=2,λ2=2/5,λ2=3. (c1) Two bell-shaped solitons and one anti-bell-shaped soliton with parameters λ1=5/2,λ2=2,λ2=1/3. (d1) Two anti-bell-shaped solitons and one bell-shaped soliton with parameters λ1=2,λ2=2/5,λ2=1/3. (a2–d2) The propagation processes for q~n at different time corresponding to (a1–d1), respectively.
(a)(b)(c)(d)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
(17)q~n=1−λ142λ12sechξ1+lnλ1,where ξ1=2lnλ1n+λ14−1/λ14+2/λ12−2λ12t and λ1>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 qn is defined by Eqn=∫−∞∞qn2dn.Table 1
Physical characteristics of the one-soliton solution.
SolitonAmplitudeWidthVelocityWave numberPrimary phaseEnergyqn1−λ14/2λ121/2lnλ1λ18−2λ16+2λ12−1/2λ14lnλ12lnλ1lnλ1λ14−12/4λ14∣lnλ1∣From the above analysis, we can find thatqn is the bell-shaped bright one-soliton when 0<λ1<1 and anti-bell-shaped one-soliton when λ1>1. When λ1=1/2 and λ1=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 one-soliton 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∈−0.5,0.5, 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∈−0.5,0.5, that is to say that the numerical results in Figure 2 show that the evolution of the exact one-soliton solution is robust against a small noise.
(II)
whenN=2, let λ=λii=1,2; we can get the two-fold exact solution from the transformation (10) as(18)q~n=bn+11,where bn1=Δbn1/Δn, in which
(19)Δn=λ12ϕnλ1ϕnλ1λ13ψnλ1λ1ψnλ1λ22ϕnλ2ϕnλ2λ23ψnλ2λ2ψnλ2λ12ψnλ1λ14ψnλ1−λ1ϕnλ1−λ13ϕnλ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−λ23ϕnλ2,Δbn1=λ12ϕnλ1ϕnλ1λ13ψnλ1−λ14ϕnλ1λ22ϕnλ2ϕnλ2λ23ψnλ2−λ24ϕnλ2λ12ψnλ1λ14ψnλ1−λ1ϕnλ1−ψnλ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−ψnλ2,while bn+11 is obtained from bn1 by replacing n with n+1. For the sake of analysis, solution (18) can be rewritten as
(20)q~n=λ12λ22−1λ12−λ22λ121−λ24coshη1+lnλ1−λ221−λ14coshη2+lnλ2λ12λ22λ12−λ222coshη1+lnλ1+η2+lnλ2+λ12λ22−12coshη1+lnλ1−η2−lnλ2−1−λ141−λ24,where ηi=2lnλin+λi4−1/λi4+2/λi2−2λi2t,i=1,2, in which λi>0 is a spectrum parameter.According to the ideas of [18, 43–45], we perform asymptotic analysis of solution (20), from which we can easily work out the limit state expressions of solution q~n before and after the interaction. And the detailed physical quantities are listed in Table 2.Table 2
Physical characteristics of the two-soliton solution.
SolitonsAmplitudesWidthsVelocitiesWave numberPrimary phasesEnergiesq1−λ14−1/2λ121/2lnλ1λ18−2λ16+2λ12−1/2λ14lnλ12lnλ1lnλ1λ12−λ22/λ12λ22−1λ14−12/4λ14lnλ1q2−1−λ24/2λ221/2lnλ2λ28−2λ26+2λ22−1/2λ24lnλ22lnλ2lnλ2λ12λ22−1/λ12−λ22λ24−12/4λ24lnλ2q1+λ14−1/2λ121/2lnλ1λ18−2λ16+2λ12−1/2λ14lnλ12lnλ1lnλ2λ12λ22−1/λ12−λ22λ14−12/4λ14lnλ1q2+1−λ24/2λ221/2lnλ2λ28−2λ26+2λ22−1/2λ24lnλ22lnλ2lnλ1λ12−λ22/λ12λ22−1λ24−12/4λ24lnλ2Before the interactionst⟶−∞,
(21)q~n⟶q1−=λ14−12λ12sechη1+lnλ1+lnλ12−λ22λ12λ22−1,η1~0,η2⟶+∞,q~n⟶q2−=−λ24−12λ22sechη2+lnλ2−lnλ12−λ22λ12λ22−1,η2~0,η1⟶+∞,where q1−, q2− are the asymptotic state expressions of q~n before the interaction.After the interactionst⟶+∞,
(22)q~n⟶q1+=λ14−12λ12sechη1+lnλ1−lnλ12−λ22λ12λ22−1,η1~0,η2⟶+∞,q~n⟶q2+=−λ24−12λ22sechη2+lnλ2+lnλ12−λ22λ12λ22−1,η2~0,η1⟶+∞,where q1+, q2+ are the asymptotic state expressions of q~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 ofqn remain unchanged before and after the interactions; (ii) after the soliton interactions, the phase shifts of two solitons are −2lnλ12−λ22/λ12λ22−1 and 2lnλ12−λ22/λ12λ22−1; and (iii) the soliton width, wave number, and wave shapes are related to two spectrum parameters λii=1,2, and the relationship between shapes and parameters are shown in Table 3.Table 3
The relationship between parameters and shapes of two solitons.
SolitonParametersShapesqnλ1>1,0<λ2<1 and λ1>1/λ2BS-BS0<λ1<1,λ2>1 and λ2>1/λ1BS-BSλ1,λ2>1 or 0<λ1,λ2<1BS-ABS0<λ1<1,λ2>1 and λ2<1/λ1ABS-ABSλ1>1,0<λ2<1 and λ1<1/λ2ABS-ABSBS stands for bell-shaped soliton, while ABS stands for anti-bell-shaped soliton.The corresponding evolution plots of solution (18) or (20) can be also elaborated clearly. When the parameters λ1=2 and λ2=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) and 3(a2); when we choose the parameters λ1=5/2 and λ2=1/2, the overtaking elastic interactions between two unidirectional bell-shaped solitons on the vanishing background are shown in Figures 3(b1) and 3(b2); in Figures 3(c1) and 3(c2), we can also see the overtaking elastic interactions between unidirectional bell-shaped soliton and anti-bell-shaped soliton on the vanishing background when λ1=2 and λ2=3.Comparing Figures3(c1) and 3(c2) with Figures 3(a1)–3(b2), we find that the soliton amplitude in Figures 3(c1) 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 have q~n=1−1/λ1λ2λ1+λ2. As λ1λ2 approaches 1, the value of q~n is small. However, when both λ1 and λ2 are greater than 1, the value of q~n is larger. Therefore, we can infer that the choice of parameters λ1 and λ2 determines the amplitude, shape, and energy of the soliton.
(III)
whenN=3, let λ=λii=1,2,3; we can get the threefold exact solution from the transformation (10):(23)q~n=bn+11,where bn1=Δbn1/Δn, in which
(24)Δn=λ14ϕnλ1λ12ϕnλ1ϕnλ1λ15ψnλ1λ13ψnλ1λ1ψnλ1λ24ϕnλ2λ22ϕnλ2ϕnλ2λ25ψnλ2λ23ψnλ2λ2ψnλ2λ34ϕnλ3λ32ϕnλ3ϕnλ3λ35ψnλ3λ33ψnλ3λ3ψnλ3λ12ψnλ1λ14ψnλ1λ16ψnλ1−λ1ϕnλ1−λ13ϕnλ1−λ15ϕnλ1λ22ψnλ2λ24ψnλ2λ26ψnλ2−λ2ϕnλ2−λ23ϕnλ2−λ25ϕnλ2λ32ψnλ3λ34ψnλ3λ36ψnλ3−λ3ϕnλ3−λ33ϕnλ3−λ35ϕnλ3,Δbn1=λ14ϕnλ1λ12ϕnλ1ϕnλ1λ15ψnλ1λ13ψnλ1−λ16ϕnλ1λ24ϕnλ2λ22ϕnλ2ϕnλ2λ25ψnλ2λ23ψnλ2−λ26ϕnλ2λ34ϕnλ3λ32ϕnλ3ϕnλ3λ35ψnλ3λ33ψnλ3−λ36ϕnλ3λ12ψnλ1λ14ψnλ1λ16ψnλ1−λ1ϕnλ1−λ13ϕnλ1−ψnλ1λ22ψnλ2λ24ψnλ2λ26ψnλ2−λ2ϕnλ2−λ23ϕnλ2−ψnλ2λ32ψnλ3λ34ψnλ3λ36ψnλ3−λ3ϕnλ3−λ33ϕnλ3−ψnλ3,while bn+11 is obtained from bn1 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 parametersλ1=5/2, λ2=1/2, and λ3=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 λ1=2, λ2=2/5, and λ3=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 λ1=5/2, λ2=2, and λ3=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 λ1=2, λ2=2/5, and λ3=1/3, the overtaking elastic interactions among two unidirectional anti-bell-shaped solitons and one bell-shaped 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 λ1,λ2, and λ3, so we will not do a detailed analysis here.


### 3.2. Discrete RS Solutions and Their Asymptotic Analysis

In this section, we will use the discrete generalized1,N−1-fold DT to construct RS solutions with the initial nonzero constant seed solution qn=c. Because of the change of the initial background, we must rewrite solution (14) as
(25)φn=ϕnψn=C1τ+neρ+t+C2τ−neρ−tC1τ+−λaτ+eρ1t+C2τ−−λaτ−eρ−t,where
(26)τ±=12λλ2+1±λ4−4c2+2λ2+1,ρ±=12λ4λ8−2λ6+6c4λ4−2λ2+1±λ6+2c2−1λ4+2c2−1λ2+1λ4−4c2+2λ2+1,and C1, C2 are arbitrary constants. Next, we fix the spectral parameter λ=λ1+ε2 with λ1=c+c2+1. In particular, if we take C1=−C2=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
(27)φnε2=∑k=0∞φnkε2k=φn0+φn1ε2+φn2ε4+φn3ε6+,⋯,with
(28)φn0=ϕn0ψn0=54n15160e1211/256tξ320−3ξ,φn1=ϕn1ψn1=54n1519660800e1211/256t3ξ3+1497600tξ+7168ξ+125337600t3ξ3−960ξ2+1497600tξ+109568ξ−34406400t−62259203,φn2=ϕn2ψn2,in which
(29)ϕn2=54n315e1211/256t7247757312000027ξ5+44928000t+645120ξ3+11280384000ξ2t+11214028800000t2+1751777280000t−7504658432ξ+1877055897600000t2+99792558489600t,ψn2=54n315e1211/256t7247757312000081ξ5−43200ξ4+134784000t+11151360ξ3−9289728000t+973209600ξ2+20426784768600000t2+33642086400000t2+2636513280000t−38242615296ξ+64958024908800t+1410963865600,and the rest φnjj≥3 are omitted here.
(I)
whenN=1, we can give the first-order RS solution of equation (1) by using the discrete generalized 1,0-fold DT as(30)q~n=can+10+bn+11,where an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(31)Δn=ϕn0λ1λ1ψn0λ1λ12ψn0λ1−λ1ϕn0λ1,Δbn1=ϕn0λ1−λ12ϕn0λ1λ12ψnλ1−ψn0λ1,Δan0=−λ12ϕn0λ1λ1ψn0λ1−ψn0λ1−λ1ϕn0λ1,while an+10,bn+11 are obtained from an0,bn1 by replacing n with n+1. So just for analysis purposes, let’s rewrite this solution as
(32)q~n=34−4.68753/2n+675/64t−1/22+1,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 that q~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 are2/3, 675/64, 3/2 and −1/2, respectively(iii)
After removing the background3/4 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 Figures5(a1) and 5(a2).
(II)
whenN=2, we can give the second-order RS solution of equation (1) via the generalized 1,1-fold DT as(33)q~n=can+10+bn+11,where an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(34)Δn=λ12ϕn0ϕn0λ13ψn0λ1ψn0λ12ψn0λ14ψn0−λ1ϕn0−λ13ϕn0λ12ϕn1+2λ1ϕn0ϕn1λ13ψn1+3λ12ψn0λ1ψn1+ψn0λ12ψn1+2λ1ψn0λ14ψn1+4λ13ψn0−λ1ϕn1−ϕ10−λ13ϕn1−3λ12ϕn0,Δan0=λ12ϕn0ϕn0−λ14ϕn0λ1ψn0λ12ψn0−ψn0−λ1ϕn0−λ13ϕn0λ12ϕn1+2λ1ϕn0−λ4ϕn1−4λ3ϕn0λ13ψn1+3λ12ψn0λ1ψn1+ψn0λ12ψn1+2λ1ψn0−ψn1−λ1ϕn1−ϕ10−λ13ϕn1−3λ12ϕn0,Δbn1=λ12ϕn0ϕn0λ13ψn0−λ14ϕn0λ12ψn0λ14ψn0−λ1ϕn0−ψn0λ12ϕn1+2λ1ϕn0ϕn1λ13ψn1+3λ12ψn0−λ4ϕn1−4λ3ϕn0λ12ψn1+2λ1ψn0λ14ψn1+4λ13ψn0−λ1ϕn1−ϕ10−ψn1.Figure 5
RS solutionq~n with the parameter λ1=2. (a1, a2) First-order RS; (b1, b2) second-order RS; (c1, c2) third-order RS. (a2–c2) The propagation processes for q~n at different time corresponding to (a1–c1), respectively.
(a)(b)(c)For the sake of discussion, the simplification form of solution (33) is listed as follows:
(35)q~n=34+FG,where
(36)F=−34789235097600n4+−1467670855680000t−185542587187200n3+−23219011584000000t2−5870683422720000t−429067232870400n2+−163258675200000000t3−61917364224000000t2−14595171287040000t−432932703436800n−430467210000000000t4−217678233600000000t3−106205478912000000t2−11958799564800000t−77309411328000,G=5566277615616n6+352241005363200t+44530220924928n5+9287604633600000t2+2348273369088000t+144723218006016n4+130606940160000000t3+49533891379200000t2+5218385264640000t+252337918574592n3+1033121304000000000t4+522427760640000000t3+68521883074560000t2+4435627474944000t+307382219440128n2+4358480501250000000t5+2754990144000000000t4+383113691136000000t3+9356401704960000t2+3822467206348800t+263882790666240n+7661391506103515625t6+5811307335000000000t5+749969539200000000t4−98680799232000000t3+41469345792000000t2+2783138807808000t+109951162777600.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−641275/161/3t1/3; we can calculate the limits of q~n in (35), which only gives the following one limiting state when t⟶±∞:
(37)q~n⟶q±=34−768003ζ−642+16384,t⟶±∞.From the above calculation, we can see thatqn± is the result of q~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 qn± 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)
whenN=3, we can give the discrete third-order RS solution of equation (1) via the generalized 1,2-fold DT as(38)q~n=can+10+bn+11,where an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(39)Δn=λ14ϕn0λ12ϕn0ϕn0λ15ψn0λ13ψn0λ1ψn0λ12ψn0λ14ψn0λ16ψn0−λ1ϕn0−λ13ϕn0−λ15ϕn0Δ3,1Δ3,2ϕn1Δ3,4Δ3,5Δ3,6Δ4,1Δ4,2Δ4,3Δ4,4Δ4,5Δ4,6Δ5,1Δ5,2ϕn2Δ5,4Δ5,5Δ5,6Δ6,1Δ6,2Δ6,3Δ6,4Δ6,5Δ6,6,Δan0=λ14ϕn0λ12ϕn0−λ16ϕn0λ15ψn0λ13ψn0λ1ψn0λ12ψn0λ14ψn0−ψn0−λ1ϕn0−λ13ϕn0−λ15ϕn0Δ3,1Δ3,2−λ16ϕn1−6λ15ϕn0Δ3,4Δ3,5Δ3,6Δ4,1Δ4,2−ψn1Δ4,4Δ4,5Δ4,6Δ5,1Δ5,2−λ16ϕn2−6λ15ϕn1−15λ14ϕn0Δ5,4Δ5,5Δ5,6Δ6,1Δ6,2−ψn2Δ6,4Δ6,5Δ6,6,Δbn1=λ14ϕn0λ12ϕn0ϕn0λ15ψn0λ13ψn0−λ16ϕn0λ12ψn0λ14ψn0λ16ψn0−λ1ϕn0−λ13ϕn0−ψn0Δ3,1Δ3,2ϕn1Δ3,4Δ3,5−λ16ϕn1−6λ15ϕn0Δ4,1Δ4,2Δ4,3Δ4,4Δ4,5−ψn1Δ5,1Δ5,2ϕn2Δ5,4Δ5,5−λ16ϕn2−6λ15ϕn1−15λ14ϕn0Δ6,1Δ6,2Δ6,3Δ6,4Δ6,5−ψn2,with
(40)Δ3,1=λ14ϕn1+4λ13ϕn0,Δ3,2=λ12ϕn1+2λ1ϕn0,Δ3,4=λ15ψn1+5λ14ψn0,Δ3,5=λ13ψn1+3λ12ψn0,Δ3,6=λ1ψn1+ψn0,Δ4,1=λ12ψn1+2λ1ψn0,Δ4,2=λ14ψn1+4λ13ψn0,Δ4,3=λ16ψn1+6λ15ψn0,Δ4,4=−λ1ϕn1−ϕ10,Δ4,5=−λ13ϕn1−3λ12ϕn0,Δ4,6=−λ15ϕn1−5λ14ϕn0,Δ5,1=λ14ϕn2+4λ13ϕn1+6λ12ϕn0,Δ5,2=λ12ϕn2+2λ1ϕn1+ϕn0,Δ5,4=λ15ψn2+5λ14ψn1+10λ13ψn0,Δ5,5=λ13ψn2+3λ12ψn1+3λ1ψn0,Δ5,6=λ1ψn2+ψn1,Δ6,1=λ12ψn2+2λ1ψn1+ψn0,Δ6,2=λ14ψn2+4λ13ψn1+6λ12ψn0,Δ6,3=λ16ψn2+6λ15ψn1+15λ14ψn0,Δ6,4=−λ1ϕn2−ϕn1,Δ6,5=−λ13ϕn2−3λ12ϕn1−3λ1ϕn0,Δ6,6=−λ15ϕn2−5λ14ϕn1−10λ13ϕn0.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 ζ1=64n+675t−646375/32+38255/321/3t1/3, ζ2=64n+675t−646375/32+38255/321/3t1/3, and b=6375/32+38255/321/3−6375/32+38255/321/3; then, it turns out that solutions q~n have two different asymptotic states when ∣t∣⟶∞:
(i)
Ifζ1=64n+675t−646375/32+38255/321/3t1/3=O1, from ζ2=ζ1−bt1/3, we can get ζ2⟶±∞ as t⟶∓∞; then, calculating the limit of solution q~n in (38) gives the following asymptotic state expression in the form(41)q~n⟶q1±=34−768003ζ1−642+16384,bt13⟶±∞.(ii)
Ifζ2=64n+675t−646375/32+38255/321/3t1/3=O1, from ζ1=ζ2+bt1/3, we can get ζ1⟶±∞ as t⟶±∞; then, calculating the limit of solution q~n in (38) gives the following asymptotic state expression as(42)q~n⟶q2±=34−768003ζ2−642+16384,bt13⟶±∞.From the above calculation, we can see thatq1± and q2± are the two results of q~n when t approaches infinity and 3ζ1−64=0 and 3ζ2−64=0 are also the two center trajectories of solution q~n. Meanwhile, the form of the third-order RS solution at infinity clearly shows that q1± and q2± reach the minimum along curves 3ζ1−64=0 and 3ζ2−64=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 interesting 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. Figure6 shows first-order 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∈−0.5,0.5, 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 the first-order RS is more sensitive to a very small noise than the usual one-soliton solution. So, we can conclude that the evolution behavior of the first-order RS solution is unstable even at a very small noise, that is to say that the RS solution is weaker than the US against small noise. This phenomenon also exists for higher-order RS solutions, which will not be discussed here.Figure 6
One-order RS (32) with the same parameters as in Figure 5 (a) RS solutions; (b) time evolutions without small noise; (c) time evolutions with 0.01% noise; (d) time evolutions with 0.05% noise.
(a)(b)(c)(d)Finally, whenN≥4, we also discuss higher-order RS solutions, which are very complicated, and we will not discuss them here. Some mathematical features of rational solution q~n for equation (1) are summarized in Table 4. It is worth noting that the first column in Table 4 means the order number of RS solutions and the second column represents the background levels of RS solutions, while the third and fourth columns exhibit the highest powers in the numerator and denominator of the polynomials involved in each RS solution, respectively.Table 4
Main mathematical features of rational solutionsq~n of order N.
NBackgroundHighest powers in the numeratorHighest powers in the denominator13/42223/46633/41212…………N3/4NN+1NN+1


### 3.3. Discrete Mixed Soliton Solutions and Their Asymptotic Analysis

In the previous contents, we have used the discreteN-fold DT with N spectral parameters to explore the US solutions and then used discrete generalized 1,N−1-fold DT with only one spectral parameter to explore RS solutions. In this section, we will use the discrete generalized 2,N−2-fold DT to construct mixed soliton solutions of equation (1) with its initial nonzero solution qn=a. In the following, we only discuss the case N=2. Besides, we also will use symbolic computation and asymptotic analysis to discuss the interaction phenomena of US and RS. When N=2, we fix the spectral parameters in equation (25) as λ1=λ1+ε2 and λ2=3 with λ1=c+c2+1. In particular, we will get the same form of Taylor series (27) if we take c=3/4 and expand the vector function φn in equation (25). Besides, when we choose C1=C2=1 in equation (25), the mixed interaction solution of the usual one-soliton and first-order RS of equation (1) can be given as
(43)q~n=can+10+bn+11,wherean0 and bn1 can be determined by the following system:
(44)Tn0λ1φn0λ1=0,Tnλ2φnλ2=0,with an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(45)Δn=λ12ϕn0λ1ϕn0λ1λ13ψn0λ1λ1ψn0λ1λ22ϕnλ2ψnλ2λ23ψnλ2λ2ψnλ2λ12ψn0λ1λ14ψn0λ1−λ1ϕn0λ1−λ13ϕn0λ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−λ23ϕnλ2,Δan0=λ12ϕn0λ1−λ14ϕn0λ1λ13ψn0λ1λ1ψn0λ1λ22ϕnλ2−λ24ϕnλ2λ23ψnλ2λ2ψnλ2λ12ψn0λ1−ψn0λ1−λ1ϕn0λ1−λ13ϕn0λ1λ22ψnλ2−ψnλ2−λ2ϕnλ2−λ23ϕnλ2,Δbn1=λ12ϕn0λ1ϕn0λ1λ13ψn0λ1−λ14ϕn0λ1λ22ϕnλ2ψnλ2λ23ψnλ2−λ24ϕnλ2λ12ψn0λ1λ14ψn0λ1−λ1ϕn0λ1−ψn0λ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−ψnλ2.With the aid of symbolic computation, solution (43) of equation (1) can be exactly expressed as
(46)q~n=34+FG,where
(47)F=−86016005759−20079n7+294e−14825t7/648+86016007−2945759+20079ne14825t7/648+135475200·25nn2+67532nt+83n+4556254096t2+2258t+361126,G=41287685759−20079n7n2+6757nt32+2307n21+4556257t24096+258757t224+14079+29n24+19575nt128+82n3+13213125t216384+9225t32+273163e−14825t7/648−41287685759+20079n7n2+6757nt32+2307n21+4556257t24096+258757t224+14079−29n24−19575nt128−82n3−13213125t216384−9225t32−273163e14825t7/648+41803776·25nn2+67532nt+83n+455625t24096+225t8−11663.Before that, we have analyzed the physical shapes and properties of US and RS solutions by using asymptotic analysis. Next, we will also discuss the mixed soliton solution before and after the interaction via asymptotic analysis. Different from the previous discussion, we make appropriate variable substitution and deformation for the mixed solution in solution (43) or (46). For convenience, let ς1=ln575+207/225n+14825t7/648t and ς2=64n+675t; then, it turns out that solution q~n has the following different asymptotic state expression when ∣t∣⟶∞:
(i)
Ifς1=O1, we can get ς2⟶±∞ as t⟶±∞; then, calculating the limit of solution q~n in (43) or (46) gives the following asymptotic expressions in the form:(i)
Before and after interactionst⟶±∞(48)q~n⟶q1±=34+17572coshς1+ln29−47/27+54,ς1⟶0,ς2⟶±∞.From which, we can see that the asymptotic expressionq1± displays the US structure which reaches the maximum 77/36 along the line ς1+ln29−47/27=0. More physical characteristics of the US structure are listed in Table 5.
(ii)
Ifς2=O1, we can get ς1⟶±∞ as t⟶±∞; then, calculating the limit of solution q~n in (43) or (46) gives the following asymptotic expressions in the form(iii)
Before the interactiont⟶−∞,(49)q~n⟶q2−=34−53760063ς2+46087+10752/1262+114688(iv)
After the interactiont⟶+∞,(50)q~n⟶q2+=34−53760063ς2−46087+10752/1262+114688.Table 5
Physical characteristics of mixed solution.
SolitonsAmplitudesWidthsVelocitiesWave numberPrimary phasesEnergiesq1±77/36225/575+207741257/8280+2887575+207/225ln29−47/27E2/E1q2−63/161/1927675/641927768+25671875π/256q2+63/161/1927675/641927768−25671875π/256E2=17868757−3045000arctan37+20300007−8338750 and E1=32447−292ln575−207/1457.According to the above analysis, we find that mixed soliton solution (46) possesses the following propagation characteristics and interaction properties: (i) the amplitudes, velocities, and energies of the mixed US-RS solution remain unchanged before and after the interactions; (ii) after the interactions of US and RS, the phase shift of US is zero, whereas the phase shift of RS is −5127, that is to say that, in the mixed soliton interaction, only RS has a phase shift but US has no phase shift; (iii) the asymptotic expression q2± displays that the RS structure reaches the minimum 63/16 along the two lines ς2∓46087+10752/126=0; and (iv) the relevant physical quantities of mixed soliton solution are listed in Table 5.For solution (43) or (46), we can see that mixed soliton solution q~n has three center trajectories and US is along one trajectory ς1+ln27/29+47=0, whereas RS is along two trajectories ς2∓46087∓10752/126=0. The structures of mixed soliton solutions are shown in Figure 7, which is consistent with our analysis above.Figure 7
Mixed interaction between usual one-soliton and first-order RS of solutionq~n in (43) or (46). (a) Mixed interaction solution of one ADS and first-order DRS with parameters λ1=2,λ2=3,C1=−C2=1. (c) Mixed interaction solution of one DS and first-order DRS with parameters λ1=2,λ2=3,C1=C2=1. (b, d) The propagation processes for q~n at t=−1 (solid line), t=0 (longdash line), and t=1 (dashdot line) corresponding to (a) and (c). The phrase DS stands for the usual dark soliton; the phrase ADS stands for the usual antidark soliton; the phrase DRS stands for dark one-order RS.
(a)(b)(c)(d)


## 3.1. UsualN-Soliton Solutions and Their Asymptotic Analysis

We can get usual soliton solutions from the vanishing background by usingN-fold DT, namely, in the condition of m=N in discrete generalized m,N−m-fold DT. Substituting the initial solutions qn=0 into Lax pair (2) can give its basic solution:
(14)φn=ϕnψn=λneλ4−2λ2tλ−ne1λ4−2λ2t.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)
whenN=1, let λ=λ1; based on the N-fold DT, we can get the onefold exact solution as(15)q~n=bn+11,where bn1=Δbn1/Δn, in which
(16)Δn=ϕnλ1λ1ψnλ1λ12ψnλ1−λ1ϕnλ1,Δbn1=ϕnλ1−λ12ϕnλ1λ12ψnλ1−ψnλ1,while bn+11 is obtained from bn1 by replacing n with n+1.Figure 1
One-soliton solutionq~n in (15) or (17). (a1) Bell-shaped one-soliton solution with the parameter λ1=1/2. (b1) Anti-bell-shaped one-soliton solution with the parameter λ1=2. (a2, b2) The propagation processes for q~n at t=−3 (dash-dot line), t=0 (long dashed line), and t=3 (solid line).
(a)(b)Figure 2
One-soliton solution (17) with the same parameter as in Figure 1. (a) Exact solution. (b) Time evolution without a small noise. (c) Time evolution with a 2% noise. (d) Time evolution with a 5% noise.
(a)(b)(c)(d)Figure 3
Two-soliton solutionq~n in (18) or (20). (a1) Two anti-bell-shaped solitons with parameters λ1=2,λ2=1/3. (b1) Two bell-shaped solitons with parameters λ1=5/2,λ2=1/2. (c1) Bell-shaped soliton and anti-bell-shaped soliton with parameters λ1=2 and λ2=3. (a2–c2) The propagation processes for q~n at different time corresponding to (a1–c1), respectively.
(a)(b)(c)Figure 4
Three-soliton solutionq~n in (23). (a1) Bell-shaped three solitons with parameters λ1=5/2,λ2=1/2,λ2=1/3. (b1) Anti-bell-shaped three solitons with parameters λ1=2,λ2=2/5,λ2=3. (c1) Two bell-shaped solitons and one anti-bell-shaped soliton with parameters λ1=5/2,λ2=2,λ2=1/3. (d1) Two anti-bell-shaped solitons and one bell-shaped soliton with parameters λ1=2,λ2=2/5,λ2=1/3. (a2–d2) The propagation processes for q~n at different time corresponding to (a1–d1), respectively.
(a)(b)(c)(d)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
(17)q~n=1−λ142λ12sechξ1+lnλ1,where ξ1=2lnλ1n+λ14−1/λ14+2/λ12−2λ12t and λ1>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 qn is defined by Eqn=∫−∞∞qn2dn.Table 1
Physical characteristics of the one-soliton solution.
SolitonAmplitudeWidthVelocityWave numberPrimary phaseEnergyqn1−λ14/2λ121/2lnλ1λ18−2λ16+2λ12−1/2λ14lnλ12lnλ1lnλ1λ14−12/4λ14∣lnλ1∣From the above analysis, we can find thatqn is the bell-shaped bright one-soliton when 0<λ1<1 and anti-bell-shaped one-soliton when λ1>1. When λ1=1/2 and λ1=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 one-soliton 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∈−0.5,0.5, 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∈−0.5,0.5, that is to say that the numerical results in Figure 2 show that the evolution of the exact one-soliton solution is robust against a small noise.
(II)
whenN=2, let λ=λii=1,2; we can get the two-fold exact solution from the transformation (10) as(18)q~n=bn+11,where bn1=Δbn1/Δn, in which
(19)Δn=λ12ϕnλ1ϕnλ1λ13ψnλ1λ1ψnλ1λ22ϕnλ2ϕnλ2λ23ψnλ2λ2ψnλ2λ12ψnλ1λ14ψnλ1−λ1ϕnλ1−λ13ϕnλ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−λ23ϕnλ2,Δbn1=λ12ϕnλ1ϕnλ1λ13ψnλ1−λ14ϕnλ1λ22ϕnλ2ϕnλ2λ23ψnλ2−λ24ϕnλ2λ12ψnλ1λ14ψnλ1−λ1ϕnλ1−ψnλ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−ψnλ2,while bn+11 is obtained from bn1 by replacing n with n+1. For the sake of analysis, solution (18) can be rewritten as
(20)q~n=λ12λ22−1λ12−λ22λ121−λ24coshη1+lnλ1−λ221−λ14coshη2+lnλ2λ12λ22λ12−λ222coshη1+lnλ1+η2+lnλ2+λ12λ22−12coshη1+lnλ1−η2−lnλ2−1−λ141−λ24,where ηi=2lnλin+λi4−1/λi4+2/λi2−2λi2t,i=1,2, in which λi>0 is a spectrum parameter.According to the ideas of [18, 43–45], we perform asymptotic analysis of solution (20), from which we can easily work out the limit state expressions of solution q~n before and after the interaction. And the detailed physical quantities are listed in Table 2.Table 2
Physical characteristics of the two-soliton solution.
SolitonsAmplitudesWidthsVelocitiesWave numberPrimary phasesEnergiesq1−λ14−1/2λ121/2lnλ1λ18−2λ16+2λ12−1/2λ14lnλ12lnλ1lnλ1λ12−λ22/λ12λ22−1λ14−12/4λ14lnλ1q2−1−λ24/2λ221/2lnλ2λ28−2λ26+2λ22−1/2λ24lnλ22lnλ2lnλ2λ12λ22−1/λ12−λ22λ24−12/4λ24lnλ2q1+λ14−1/2λ121/2lnλ1λ18−2λ16+2λ12−1/2λ14lnλ12lnλ1lnλ2λ12λ22−1/λ12−λ22λ14−12/4λ14lnλ1q2+1−λ24/2λ221/2lnλ2λ28−2λ26+2λ22−1/2λ24lnλ22lnλ2lnλ1λ12−λ22/λ12λ22−1λ24−12/4λ24lnλ2Before the interactionst⟶−∞,
(21)q~n⟶q1−=λ14−12λ12sechη1+lnλ1+lnλ12−λ22λ12λ22−1,η1~0,η2⟶+∞,q~n⟶q2−=−λ24−12λ22sechη2+lnλ2−lnλ12−λ22λ12λ22−1,η2~0,η1⟶+∞,where q1−, q2− are the asymptotic state expressions of q~n before the interaction.After the interactionst⟶+∞,
(22)q~n⟶q1+=λ14−12λ12sechη1+lnλ1−lnλ12−λ22λ12λ22−1,η1~0,η2⟶+∞,q~n⟶q2+=−λ24−12λ22sechη2+lnλ2+lnλ12−λ22λ12λ22−1,η2~0,η1⟶+∞,where q1+, q2+ are the asymptotic state expressions of q~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 ofqn remain unchanged before and after the interactions; (ii) after the soliton interactions, the phase shifts of two solitons are −2lnλ12−λ22/λ12λ22−1 and 2lnλ12−λ22/λ12λ22−1; and (iii) the soliton width, wave number, and wave shapes are related to two spectrum parameters λii=1,2, and the relationship between shapes and parameters are shown in Table 3.Table 3
The relationship between parameters and shapes of two solitons.
SolitonParametersShapesqnλ1>1,0<λ2<1 and λ1>1/λ2BS-BS0<λ1<1,λ2>1 and λ2>1/λ1BS-BSλ1,λ2>1 or 0<λ1,λ2<1BS-ABS0<λ1<1,λ2>1 and λ2<1/λ1ABS-ABSλ1>1,0<λ2<1 and λ1<1/λ2ABS-ABSBS stands for bell-shaped soliton, while ABS stands for anti-bell-shaped soliton.The corresponding evolution plots of solution (18) or (20) can be also elaborated clearly. When the parameters λ1=2 and λ2=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) and 3(a2); when we choose the parameters λ1=5/2 and λ2=1/2, the overtaking elastic interactions between two unidirectional bell-shaped solitons on the vanishing background are shown in Figures 3(b1) and 3(b2); in Figures 3(c1) and 3(c2), we can also see the overtaking elastic interactions between unidirectional bell-shaped soliton and anti-bell-shaped soliton on the vanishing background when λ1=2 and λ2=3.Comparing Figures3(c1) and 3(c2) with Figures 3(a1)–3(b2), we find that the soliton amplitude in Figures 3(c1) 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 have q~n=1−1/λ1λ2λ1+λ2. As λ1λ2 approaches 1, the value of q~n is small. However, when both λ1 and λ2 are greater than 1, the value of q~n is larger. Therefore, we can infer that the choice of parameters λ1 and λ2 determines the amplitude, shape, and energy of the soliton.
(III)
whenN=3, let λ=λii=1,2,3; we can get the threefold exact solution from the transformation (10):(23)q~n=bn+11,where bn1=Δbn1/Δn, in which
(24)Δn=λ14ϕnλ1λ12ϕnλ1ϕnλ1λ15ψnλ1λ13ψnλ1λ1ψnλ1λ24ϕnλ2λ22ϕnλ2ϕnλ2λ25ψnλ2λ23ψnλ2λ2ψnλ2λ34ϕnλ3λ32ϕnλ3ϕnλ3λ35ψnλ3λ33ψnλ3λ3ψnλ3λ12ψnλ1λ14ψnλ1λ16ψnλ1−λ1ϕnλ1−λ13ϕnλ1−λ15ϕnλ1λ22ψnλ2λ24ψnλ2λ26ψnλ2−λ2ϕnλ2−λ23ϕnλ2−λ25ϕnλ2λ32ψnλ3λ34ψnλ3λ36ψnλ3−λ3ϕnλ3−λ33ϕnλ3−λ35ϕnλ3,Δbn1=λ14ϕnλ1λ12ϕnλ1ϕnλ1λ15ψnλ1λ13ψnλ1−λ16ϕnλ1λ24ϕnλ2λ22ϕnλ2ϕnλ2λ25ψnλ2λ23ψnλ2−λ26ϕnλ2λ34ϕnλ3λ32ϕnλ3ϕnλ3λ35ψnλ3λ33ψnλ3−λ36ϕnλ3λ12ψnλ1λ14ψnλ1λ16ψnλ1−λ1ϕnλ1−λ13ϕnλ1−ψnλ1λ22ψnλ2λ24ψnλ2λ26ψnλ2−λ2ϕnλ2−λ23ϕnλ2−ψnλ2λ32ψnλ3λ34ψnλ3λ36ψnλ3−λ3ϕnλ3−λ33ϕnλ3−ψnλ3,while bn+11 is obtained from bn1 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 parametersλ1=5/2, λ2=1/2, and λ3=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 λ1=2, λ2=2/5, and λ3=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 λ1=5/2, λ2=2, and λ3=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 λ1=2, λ2=2/5, and λ3=1/3, the overtaking elastic interactions among two unidirectional anti-bell-shaped solitons and one bell-shaped 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 λ1,λ2, and λ3, so we will not do a detailed analysis here.


## 3.2. Discrete RS Solutions and Their Asymptotic Analysis

In this section, we will use the discrete generalized1,N−1-fold DT to construct RS solutions with the initial nonzero constant seed solution qn=c. Because of the change of the initial background, we must rewrite solution (14) as
(25)φn=ϕnψn=C1τ+neρ+t+C2τ−neρ−tC1τ+−λaτ+eρ1t+C2τ−−λaτ−eρ−t,where
(26)τ±=12λλ2+1±λ4−4c2+2λ2+1,ρ±=12λ4λ8−2λ6+6c4λ4−2λ2+1±λ6+2c2−1λ4+2c2−1λ2+1λ4−4c2+2λ2+1,and C1, C2 are arbitrary constants. Next, we fix the spectral parameter λ=λ1+ε2 with λ1=c+c2+1. In particular, if we take C1=−C2=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
(27)φnε2=∑k=0∞φnkε2k=φn0+φn1ε2+φn2ε4+φn3ε6+,⋯,with
(28)φn0=ϕn0ψn0=54n15160e1211/256tξ320−3ξ,φn1=ϕn1ψn1=54n1519660800e1211/256t3ξ3+1497600tξ+7168ξ+125337600t3ξ3−960ξ2+1497600tξ+109568ξ−34406400t−62259203,φn2=ϕn2ψn2,in which
(29)ϕn2=54n315e1211/256t7247757312000027ξ5+44928000t+645120ξ3+11280384000ξ2t+11214028800000t2+1751777280000t−7504658432ξ+1877055897600000t2+99792558489600t,ψn2=54n315e1211/256t7247757312000081ξ5−43200ξ4+134784000t+11151360ξ3−9289728000t+973209600ξ2+20426784768600000t2+33642086400000t2+2636513280000t−38242615296ξ+64958024908800t+1410963865600,and the rest φnjj≥3 are omitted here.
(I)
whenN=1, we can give the first-order RS solution of equation (1) by using the discrete generalized 1,0-fold DT as(30)q~n=can+10+bn+11,where an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(31)Δn=ϕn0λ1λ1ψn0λ1λ12ψn0λ1−λ1ϕn0λ1,Δbn1=ϕn0λ1−λ12ϕn0λ1λ12ψnλ1−ψn0λ1,Δan0=−λ12ϕn0λ1λ1ψn0λ1−ψn0λ1−λ1ϕn0λ1,while an+10,bn+11 are obtained from an0,bn1 by replacing n with n+1. So just for analysis purposes, let’s rewrite this solution as
(32)q~n=34−4.68753/2n+675/64t−1/22+1,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 that q~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 are2/3, 675/64, 3/2 and −1/2, respectively(iii)
After removing the background3/4 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 Figures5(a1) and 5(a2).
(II)
whenN=2, we can give the second-order RS solution of equation (1) via the generalized 1,1-fold DT as(33)q~n=can+10+bn+11,where an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(34)Δn=λ12ϕn0ϕn0λ13ψn0λ1ψn0λ12ψn0λ14ψn0−λ1ϕn0−λ13ϕn0λ12ϕn1+2λ1ϕn0ϕn1λ13ψn1+3λ12ψn0λ1ψn1+ψn0λ12ψn1+2λ1ψn0λ14ψn1+4λ13ψn0−λ1ϕn1−ϕ10−λ13ϕn1−3λ12ϕn0,Δan0=λ12ϕn0ϕn0−λ14ϕn0λ1ψn0λ12ψn0−ψn0−λ1ϕn0−λ13ϕn0λ12ϕn1+2λ1ϕn0−λ4ϕn1−4λ3ϕn0λ13ψn1+3λ12ψn0λ1ψn1+ψn0λ12ψn1+2λ1ψn0−ψn1−λ1ϕn1−ϕ10−λ13ϕn1−3λ12ϕn0,Δbn1=λ12ϕn0ϕn0λ13ψn0−λ14ϕn0λ12ψn0λ14ψn0−λ1ϕn0−ψn0λ12ϕn1+2λ1ϕn0ϕn1λ13ψn1+3λ12ψn0−λ4ϕn1−4λ3ϕn0λ12ψn1+2λ1ψn0λ14ψn1+4λ13ψn0−λ1ϕn1−ϕ10−ψn1.Figure 5
RS solutionq~n with the parameter λ1=2. (a1, a2) First-order RS; (b1, b2) second-order RS; (c1, c2) third-order RS. (a2–c2) The propagation processes for q~n at different time corresponding to (a1–c1), respectively.
(a)(b)(c)For the sake of discussion, the simplification form of solution (33) is listed as follows:
(35)q~n=34+FG,where
(36)F=−34789235097600n4+−1467670855680000t−185542587187200n3+−23219011584000000t2−5870683422720000t−429067232870400n2+−163258675200000000t3−61917364224000000t2−14595171287040000t−432932703436800n−430467210000000000t4−217678233600000000t3−106205478912000000t2−11958799564800000t−77309411328000,G=5566277615616n6+352241005363200t+44530220924928n5+9287604633600000t2+2348273369088000t+144723218006016n4+130606940160000000t3+49533891379200000t2+5218385264640000t+252337918574592n3+1033121304000000000t4+522427760640000000t3+68521883074560000t2+4435627474944000t+307382219440128n2+4358480501250000000t5+2754990144000000000t4+383113691136000000t3+9356401704960000t2+3822467206348800t+263882790666240n+7661391506103515625t6+5811307335000000000t5+749969539200000000t4−98680799232000000t3+41469345792000000t2+2783138807808000t+109951162777600.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−641275/161/3t1/3; we can calculate the limits of q~n in (35), which only gives the following one limiting state when t⟶±∞:
(37)q~n⟶q±=34−768003ζ−642+16384,t⟶±∞.From the above calculation, we can see thatqn± is the result of q~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 qn± 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)
whenN=3, we can give the discrete third-order RS solution of equation (1) via the generalized 1,2-fold DT as(38)q~n=can+10+bn+11,where an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(39)Δn=λ14ϕn0λ12ϕn0ϕn0λ15ψn0λ13ψn0λ1ψn0λ12ψn0λ14ψn0λ16ψn0−λ1ϕn0−λ13ϕn0−λ15ϕn0Δ3,1Δ3,2ϕn1Δ3,4Δ3,5Δ3,6Δ4,1Δ4,2Δ4,3Δ4,4Δ4,5Δ4,6Δ5,1Δ5,2ϕn2Δ5,4Δ5,5Δ5,6Δ6,1Δ6,2Δ6,3Δ6,4Δ6,5Δ6,6,Δan0=λ14ϕn0λ12ϕn0−λ16ϕn0λ15ψn0λ13ψn0λ1ψn0λ12ψn0λ14ψn0−ψn0−λ1ϕn0−λ13ϕn0−λ15ϕn0Δ3,1Δ3,2−λ16ϕn1−6λ15ϕn0Δ3,4Δ3,5Δ3,6Δ4,1Δ4,2−ψn1Δ4,4Δ4,5Δ4,6Δ5,1Δ5,2−λ16ϕn2−6λ15ϕn1−15λ14ϕn0Δ5,4Δ5,5Δ5,6Δ6,1Δ6,2−ψn2Δ6,4Δ6,5Δ6,6,Δbn1=λ14ϕn0λ12ϕn0ϕn0λ15ψn0λ13ψn0−λ16ϕn0λ12ψn0λ14ψn0λ16ψn0−λ1ϕn0−λ13ϕn0−ψn0Δ3,1Δ3,2ϕn1Δ3,4Δ3,5−λ16ϕn1−6λ15ϕn0Δ4,1Δ4,2Δ4,3Δ4,4Δ4,5−ψn1Δ5,1Δ5,2ϕn2Δ5,4Δ5,5−λ16ϕn2−6λ15ϕn1−15λ14ϕn0Δ6,1Δ6,2Δ6,3Δ6,4Δ6,5−ψn2,with
(40)Δ3,1=λ14ϕn1+4λ13ϕn0,Δ3,2=λ12ϕn1+2λ1ϕn0,Δ3,4=λ15ψn1+5λ14ψn0,Δ3,5=λ13ψn1+3λ12ψn0,Δ3,6=λ1ψn1+ψn0,Δ4,1=λ12ψn1+2λ1ψn0,Δ4,2=λ14ψn1+4λ13ψn0,Δ4,3=λ16ψn1+6λ15ψn0,Δ4,4=−λ1ϕn1−ϕ10,Δ4,5=−λ13ϕn1−3λ12ϕn0,Δ4,6=−λ15ϕn1−5λ14ϕn0,Δ5,1=λ14ϕn2+4λ13ϕn1+6λ12ϕn0,Δ5,2=λ12ϕn2+2λ1ϕn1+ϕn0,Δ5,4=λ15ψn2+5λ14ψn1+10λ13ψn0,Δ5,5=λ13ψn2+3λ12ψn1+3λ1ψn0,Δ5,6=λ1ψn2+ψn1,Δ6,1=λ12ψn2+2λ1ψn1+ψn0,Δ6,2=λ14ψn2+4λ13ψn1+6λ12ψn0,Δ6,3=λ16ψn2+6λ15ψn1+15λ14ψn0,Δ6,4=−λ1ϕn2−ϕn1,Δ6,5=−λ13ϕn2−3λ12ϕn1−3λ1ϕn0,Δ6,6=−λ15ϕn2−5λ14ϕn1−10λ13ϕn0.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 ζ1=64n+675t−646375/32+38255/321/3t1/3, ζ2=64n+675t−646375/32+38255/321/3t1/3, and b=6375/32+38255/321/3−6375/32+38255/321/3; then, it turns out that solutions q~n have two different asymptotic states when ∣t∣⟶∞:
(i)
Ifζ1=64n+675t−646375/32+38255/321/3t1/3=O1, from ζ2=ζ1−bt1/3, we can get ζ2⟶±∞ as t⟶∓∞; then, calculating the limit of solution q~n in (38) gives the following asymptotic state expression in the form(41)q~n⟶q1±=34−768003ζ1−642+16384,bt13⟶±∞.(ii)
Ifζ2=64n+675t−646375/32+38255/321/3t1/3=O1, from ζ1=ζ2+bt1/3, we can get ζ1⟶±∞ as t⟶±∞; then, calculating the limit of solution q~n in (38) gives the following asymptotic state expression as(42)q~n⟶q2±=34−768003ζ2−642+16384,bt13⟶±∞.From the above calculation, we can see thatq1± and q2± are the two results of q~n when t approaches infinity and 3ζ1−64=0 and 3ζ2−64=0 are also the two center trajectories of solution q~n. Meanwhile, the form of the third-order RS solution at infinity clearly shows that q1± and q2± reach the minimum along curves 3ζ1−64=0 and 3ζ2−64=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 interesting 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. Figure6 shows first-order 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∈−0.5,0.5, 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 the first-order RS is more sensitive to a very small noise than the usual one-soliton solution. So, we can conclude that the evolution behavior of the first-order RS solution is unstable even at a very small noise, that is to say that the RS solution is weaker than the US against small noise. This phenomenon also exists for higher-order RS solutions, which will not be discussed here.Figure 6
One-order RS (32) with the same parameters as in Figure 5 (a) RS solutions; (b) time evolutions without small noise; (c) time evolutions with 0.01% noise; (d) time evolutions with 0.05% noise.
(a)(b)(c)(d)Finally, whenN≥4, we also discuss higher-order RS solutions, which are very complicated, and we will not discuss them here. Some mathematical features of rational solution q~n for equation (1) are summarized in Table 4. It is worth noting that the first column in Table 4 means the order number of RS solutions and the second column represents the background levels of RS solutions, while the third and fourth columns exhibit the highest powers in the numerator and denominator of the polynomials involved in each RS solution, respectively.Table 4
Main mathematical features of rational solutionsq~n of order N.
NBackgroundHighest powers in the numeratorHighest powers in the denominator13/42223/46633/41212…………N3/4NN+1NN+1


## 3.3. Discrete Mixed Soliton Solutions and Their Asymptotic Analysis

In the previous contents, we have used the discreteN-fold DT with N spectral parameters to explore the US solutions and then used discrete generalized 1,N−1-fold DT with only one spectral parameter to explore RS solutions. In this section, we will use the discrete generalized 2,N−2-fold DT to construct mixed soliton solutions of equation (1) with its initial nonzero solution qn=a. In the following, we only discuss the case N=2. Besides, we also will use symbolic computation and asymptotic analysis to discuss the interaction phenomena of US and RS. When N=2, we fix the spectral parameters in equation (25) as λ1=λ1+ε2 and λ2=3 with λ1=c+c2+1. In particular, we will get the same form of Taylor series (27) if we take c=3/4 and expand the vector function φn in equation (25). Besides, when we choose C1=C2=1 in equation (25), the mixed interaction solution of the usual one-soliton and first-order RS of equation (1) can be given as
(43)q~n=can+10+bn+11,wherean0 and bn1 can be determined by the following system:
(44)Tn0λ1φn0λ1=0,Tnλ2φnλ2=0,with an0=Δan0/Δn and bn1=Δbn1/Δn, in which
(45)Δn=λ12ϕn0λ1ϕn0λ1λ13ψn0λ1λ1ψn0λ1λ22ϕnλ2ψnλ2λ23ψnλ2λ2ψnλ2λ12ψn0λ1λ14ψn0λ1−λ1ϕn0λ1−λ13ϕn0λ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−λ23ϕnλ2,Δan0=λ12ϕn0λ1−λ14ϕn0λ1λ13ψn0λ1λ1ψn0λ1λ22ϕnλ2−λ24ϕnλ2λ23ψnλ2λ2ψnλ2λ12ψn0λ1−ψn0λ1−λ1ϕn0λ1−λ13ϕn0λ1λ22ψnλ2−ψnλ2−λ2ϕnλ2−λ23ϕnλ2,Δbn1=λ12ϕn0λ1ϕn0λ1λ13ψn0λ1−λ14ϕn0λ1λ22ϕnλ2ψnλ2λ23ψnλ2−λ24ϕnλ2λ12ψn0λ1λ14ψn0λ1−λ1ϕn0λ1−ψn0λ1λ22ψnλ2λ24ψnλ2−λ2ϕnλ2−ψnλ2.With the aid of symbolic computation, solution (43) of equation (1) can be exactly expressed as
(46)q~n=34+FG,where
(47)F=−86016005759−20079n7+294e−14825t7/648+86016007−2945759+20079ne14825t7/648+135475200·25nn2+67532nt+83n+4556254096t2+2258t+361126,G=41287685759−20079n7n2+6757nt32+2307n21+4556257t24096+258757t224+14079+29n24+19575nt128+82n3+13213125t216384+9225t32+273163e−14825t7/648−41287685759+20079n7n2+6757nt32+2307n21+4556257t24096+258757t224+14079−29n24−19575nt128−82n3−13213125t216384−9225t32−273163e14825t7/648+41803776·25nn2+67532nt+83n+455625t24096+225t8−11663.Before that, we have analyzed the physical shapes and properties of US and RS solutions by using asymptotic analysis. Next, we will also discuss the mixed soliton solution before and after the interaction via asymptotic analysis. Different from the previous discussion, we make appropriate variable substitution and deformation for the mixed solution in solution (43) or (46). For convenience, let ς1=ln575+207/225n+14825t7/648t and ς2=64n+675t; then, it turns out that solution q~n has the following different asymptotic state expression when ∣t∣⟶∞:
(i)
Ifς1=O1, we can get ς2⟶±∞ as t⟶±∞; then, calculating the limit of solution q~n in (43) or (46) gives the following asymptotic expressions in the form:(i)
Before and after interactionst⟶±∞(48)q~n⟶q1±=34+17572coshς1+ln29−47/27+54,ς1⟶0,ς2⟶±∞.From which, we can see that the asymptotic expressionq1± displays the US structure which reaches the maximum 77/36 along the line ς1+ln29−47/27=0. More physical characteristics of the US structure are listed in Table 5.
(ii)
Ifς2=O1, we can get ς1⟶±∞ as t⟶±∞; then, calculating the limit of solution q~n in (43) or (46) gives the following asymptotic expressions in the form(iii)
Before the interactiont⟶−∞,(49)q~n⟶q2−=34−53760063ς2+46087+10752/1262+114688(iv)
After the interactiont⟶+∞,(50)q~n⟶q2+=34−53760063ς2−46087+10752/1262+114688.Table 5
Physical characteristics of mixed solution.
SolitonsAmplitudesWidthsVelocitiesWave numberPrimary phasesEnergiesq1±77/36225/575+207741257/8280+2887575+207/225ln29−47/27E2/E1q2−63/161/1927675/641927768+25671875π/256q2+63/161/1927675/641927768−25671875π/256E2=17868757−3045000arctan37+20300007−8338750 and E1=32447−292ln575−207/1457.According to the above analysis, we find that mixed soliton solution (46) possesses the following propagation characteristics and interaction properties: (i) the amplitudes, velocities, and energies of the mixed US-RS solution remain unchanged before and after the interactions; (ii) after the interactions of US and RS, the phase shift of US is zero, whereas the phase shift of RS is −5127, that is to say that, in the mixed soliton interaction, only RS has a phase shift but US has no phase shift; (iii) the asymptotic expression q2± displays that the RS structure reaches the minimum 63/16 along the two lines ς2∓46087+10752/126=0; and (iv) the relevant physical quantities of mixed soliton solution are listed in Table 5.For solution (43) or (46), we can see that mixed soliton solution q~n has three center trajectories and US is along one trajectory ς1+ln27/29+47=0, whereas RS is along two trajectories ς2∓46087∓10752/126=0. The structures of mixed soliton solutions are shown in Figure 7, which is consistent with our analysis above.Figure 7
Mixed interaction between usual one-soliton and first-order RS of solutionq~n in (43) or (46). (a) Mixed interaction solution of one ADS and first-order DRS with parameters λ1=2,λ2=3,C1=−C2=1. (c) Mixed interaction solution of one DS and first-order DRS with parameters λ1=2,λ2=3,C1=C2=1. (b, d) The propagation processes for q~n at t=−1 (solid line), t=0 (longdash line), and t=1 (dashdot line) corresponding to (a) and (c). The phrase DS stands for the usual dark soliton; the phrase ADS stands for the usual antidark soliton; the phrase DRS stands for dark one-order RS.
(a)(b)(c)(d)


## 4. 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 m,N−m-fold DT of equation (1) for the first time. Meanwhile, various kinds of soliton solutions such as US, RS, and their mixed interaction soliton solutions have been obtained via the generalized m,N−m-fold DT by choosing different values of the number m of spectral parameter λ. More importantly, asymptotic analysis has been used to analyze the limit state expressions of US, RS, and their mixed soliton solutions for equation (1). And the propagation and interaction structures between/among different types of soliton solutions have been discussed graphically. Besides, numerical simulations have been used to explore the dynamical behaviors of the US and RS solutions. Finally, some important physical quantities of US solutions, mathematical features of RS solutions, and physical quantities of mixed solutions to equation (1) have been summarized in 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.



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