Analytical Insights into a Generalized Semidiscrete System with Time-Varying Coefficients: Derivation, Exact Solutions, and Nonlinear Soliton Dynamics

Zhang, Sheng; Zhao, Sen; Xu, Bo

doi:https://doi.org/10.1155/2020/1543503

Complexity

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 1543503 | https://doi.org/10.1155/2020/1543503

Analytical Insights into a Generalized Semidiscrete System with Time-Varying Coefficients: Derivation, Exact Solutions, and Nonlinear Soliton Dynamics

Sheng Zhang,^1,2Sen Zhao,¹and Bo Xu^3,4

Academic Editor: Giacomo Innocenti

Received13 Sept 2019

Revised30 Dec 2019

Accepted14 Jan 2020

Published20 Feb 2020

Abstract

In this paper, a new generalized semidiscrete integrable system with time-varying coefficients is analytically studied. Firstly, the generalized semidiscrete system is derived from a semidiscrete matrix spectral problem by embedding finite time-varying coefficient functions. Secondly, exact and explicit N-soliton solutions of the semidiscrete system are obtained by using the inverse scattering analysis. Finally, three special cases when of the obtained N-soliton solutions are simulated by selecting some appropriate coefficient functions. It is shown that the time-varying coefficient functions affect the spatiotemporal structures and the propagation velocities of the obtained semidiscrete one-soliton solutions, two-soliton solutions, and three-soliton solutions.

1. Introduction

Semidiscrete systems keep some or all their spatial variables discrete while time continuous systems have an important role in simulating complex phenomena in many fields, for instance; they often arise in high energy physics as approximations of continuum models for the numerical simulation of nonlinear soliton dynamics [1]. It is Toda [2] who first derived the classical semidiscrete system—Toda lattice when the lattice with exponential interaction was considered. Like the famous Kortweg-de Vries equation, the classical Toda lattice is one of the most important completely integrable systems, and the multisoliton solutions of which have very important significance to the study of many nonlinear problems in atomic and particle physics. The semidiscrete sine-Gordon equation can build up a connection with the dynamics of circular arrays of Josephson junctions [3]. For the important role of the semidiscrete nonlinear Schrödinger equation, it is worth mentioning that this equation has been used to describe transport on a dimer [4] and beam propagation in Kerr nonlinear media [5].

Compared with differential or difference equations, it is difficult to derive or solve the nonlinear semidiscrete systems [6–15]. In [9, 11], an effective algorithm of the Jacobian elliptic function method devised by Dai and Zhang [6] is extended to solve the semidiscrete sine-Gordon equation and the semidiscrete nonlinear Schrödinger equation, respectively. Zhu’s exp-function ansatz [7] applied to the hybrid-lattice system and the lattice equation [8] is improved for N-soliton solutions of the Toda lattice equation [12]. Egorova, Michor, and Teschl [10] used the double commutation method and the inverse scattering transform to solve the Toda hierarchy on a quasiperiodic finite-gap background. In [13, 15], the mixed Toda lattice hierarchy and the variable-coefficient non-isospectral Toda lattice hierarchy are derived and solved by means of the inverse scattering transform. In 2014, Zhang and Liu [14] obtained N-soliton solutions of a (2 + 1)-dimensional variable-coefficient Toda lattice equation through Hirota’s bilinear method.

Recently, some new analytical approaches [16–25], novel nonlinear results [26–34], and soliton dynamics in inhomogeneous media [35–37] for differential equations have been reported. Generally speaking, extending the existing methods for differential equations encounters the difficulty in searching for the iterative relations of discrete variables. When the nonuniformities of boundaries or the inhomogeneities of media are taken into consideration, the systems with variable coefficients [38–43] are much better than their idealized counterparts—constant-coefficient models in describing more realistic phenomena. In 2011, Liu et al. [38] show that when such nonuniformities of boundaries and inhomogeneities of media are depicted by variable coefficients, the soliton behavior they found may be helpful to study the internal solitary waves in ocean- or atmosphere-stratified fluid. In 2017, Dai et al. [40] pointed out that the analysis and results they first reported for the Peregrine solution and breather solution of a nonlinear Schrödinger equation with variable coefficients could be expected to give new insight into higher-dimensional localized rogue waves in nonlocal media. It is shown in [39, 41–43] that the variable coefficients influence the spatial structures and the dynamical evolutions of solutions.

In this article, we analytically study a new generalized semidiscrete system of the form:where , , , , are all functions with respective to and and is a matrix operator in the following form:

In matrix operator (2), is a shift operator defined as for any integer number . With the help of the shift operator , we can conveniently determine the operator which has the inverse operator defined asorprovided rapidly as . At the same time, for , there holds the relation .

It should be noted that system (1) is not only different from the known ones [2, 6–15, 44–46]:but more general than the Toda lattice hierarchy [47]:

This is because, on the one hand, system (1) is a superposition of finite terms and on the other hand, system (1) contains some time-varying coefficient functions. Obviously, a special case of system (1) is equation (6) which concludes two semidiscrete systems:

A direct computation shows that and can rewrite system (7) as the celebrated Toda lattice, i.e., . If we set , then system (1) gives the following new semidiscrete system:which cannot be included in known system (6). In the literature, there is no research result about semidiscrete system (1).

Constructing or solving semidiscrete systems is a significant work in the theory of nonlinear lattices [2]. In 1990, Tu [48] proposed a scheme for generating hierarchies of semidiscrete integrable systems. In 2005, Yang and Xu [49] developed a direct method for constructing integrable expanding models for semidiscrete systems. In soliton theory, there are many methods like those in [50–58] besides the inverse scattering method [59] for solving nonlinear differential equations. In this paper, we shall analytically derive and solve system (1) by means of the inverse scattering analysis. Employing other methods to construct new solutions of system (1) is worthy of studying. In many research fields, more and more attention has been paid to fractional-order models [60–76]. Recently, Aslan [77–80] has successfully extended analytical methods-combined symbolic computation to solve fractional semidiscrete equations. How to extend the method used in this paper to such fractional semidiscrete equations is also worthy of studying.

The rest of this paper is organized as follows. In Section 2, we derive system (1). In Section 3, we construct the formulae of the N-soliton solutions of system (1). In Section 4, we analyze the dynamical evolutions of the obtained soliton solutions. In Section 5, we conclude this paper.

2. Derivation

To derive system (1), we consider the following semidiscrete matrix spectral problem [13, 15, 44, 47]:where the potential functions and are smooth enough and rapidly as , the spectral parameter , and , , , and are the functions of , , and to be determined later. Then, we employ spectral problem (10) and its time evolution equation (11) equipped with different isospectral parameter and functions , , , and to derive system (1).

Theorem 1. System (1) is Lax integrable, which can be derived fromby introducing different isospectral parameter and functions , , , and .

Proof. Substituting the matrixes and in equations (10) and (11) into equation (12) yieldswhich can be simplified asby introducing two operators and :In order to derive system (1), we equivalently rewrite equation (17) asand suppose that and can be expressed bywith the conditionsWe substitute equation (20) into equation (19) and compare the coefficients of , then the following equations are obtained:With the help of equations (2), (18), and (24), from equation (23), we haveSubstituting equation (25) into equation (22) yieldswhich can be rewritten as system (1) by employing the operator . Thus, system (1) is integrable in Lax’s sense. In view of equations (10) and (11), the Lax pair of system (1) can be written asby cancelling and setting and .

3. Exact Solutions

In this section, we would like to solve system (1) through the inverse scattering method. To avoid duplication, we omit here the similar direct and inverse scattering analysis [47].

Theorem 2. Suppose the time evolutions of and obey system (1), then the scattering datafor the isospectral problem (27) have the following time dependences:where and are the scattering data when and .

Proof. For spectral problem (27), there exist two pairs of Jost solutions [47]:where and are analytic for on the complex plane of , and are analytic for , and and for .
With the above preparations, we first consider the continuous spectrum. Since is also a solution of equation (27), there exist two functions and such thatLetting and using equations (13), (21), (32), and (33), from equation (34), we haveIn view of the arbitrariness of , from equation (35), we obtainThus, equation (34) is simplified asSubstituting the linear relation into equation (37) and setting yieldsfrom which we haveand then we obtainWe next consider the time dependences of the discrete scattering data. For such purpose, we take the Jost solution which satisfies the asymptotic property in equation (34) and set ; then, the linear relation is also a solution of equation (27). Thus, there exist two undetermined functions and such thatwhere and as . Therefore, and equation (41) have a simplified form as follows:Letting , here , we haveThen, equation (42) can be rewritten asWith the help of equation (43), we simplify equation (44) asMultiplying equation (45) by and then summing it, we havewhich impliesHere, the assumption that is a normalized eigenfunction, i.e., , and the following two results have been used:It is easy to see that equation (49) is obvious. For equation (48), inspired by the inner product [47], we haveSubstituting equation (50) into equation (48), then the summation of the left hand reaches zero.
In view of as and equations (21), (48), and (49), from equation (44), we then havewhich givesSolving equation (52), we can easily arrive at equation (31).

Theorem 3. Based on the scattering data (29)–(31), exact and explicit semidiscrete N-soliton solutions of the system (1) can be obtained:where is the N-th-order identity matrix, denotes the trace of a matrix, and

Proof. Supposingand substituting them into equation (27), then we can determine and hence recover and :We note here that satisfies the following Gel’fand–Levitan–Marchenko integral equation of discrete form [47]:whereTo construct N-soliton solutions of system (1), we set the reflection coefficient to zero. In this case, from (60), we haveFurther takingand substituting equations (61) and (62) into equation (58), we havewhich can be rewritten asby means of the vector .
Considering the positive definiteness of when the scattering data and are all real numbers, we have from equation (62),When , can be determined as follows [47]:Finally, from equations (57), (65), and (66) we obtain equations (53) and (54).

4. Nonlinear Soliton Dynamics

We study in this section the nonlinear soliton dynamics of the semidiscrete soliton solutions (53) and (54) in the three cases when .

When , we havewhereThen, the semidiscrete one-soliton solutions of system (1) are obtained:

We simulate the spatiotemporal structures of the semidiscrete one-soliton solutions (69) and (70) in Figures 1 and 2, where , , , , , , and . It can be seen that the bell-solitons determined by solutions (69) and (70) have very similar structures. In fact, the computer simulation hints . If we rewrite (69) and (70) to have the same numerator:then the subtraction of the denominators of and is calculated aswhich is very small when the parameters are selected as those of Figures 1 and 2. In Figure 3, we show the dynamical evolutions of bell-soliton solution (69) at three different times. We can see from Figures 1 and 3 that the bell-soliton solution first propagates in the positive direction of n-axis and then in the reverse direction. It is the soliton propagation in different directions that forms the V-shaped trajectory in Figure 1. However, if we let and all the other parameters unchanged, then the trajectory is a straight line (see Figure 4). Since the dynamical evolutions of solution (70) are very similar to the ones of solution (69), we omit them here for simplicity.

(a)

(b)

(c)

When , we havewhereWith the help of equations (73) and (74), we simulate in Figure 5, the spatiotemporal structure of the semidiscrete two-soliton solution determined by equation (53). Here, the parameters are selected as , , , , , , , , and . In Figure 6, we simulate a catching up process between the bell-shaped two-soliton solutions. In the process of chasing each other, the high soliton gradually surpasses the low soliton after turning and therefore forms a U-shaped trajectory, in which there is no complete collision. If we let and all the other parameters unchanged, then the two-soliton solution shown in Figure 6 propagates in a straight line and such process of chasing each other has not happened within the same time period.

For the nonlinear soliton dynamics of the semidiscrete three-soliton solutions of system (1), we set and havewhere

Based on equation (75), the spatiotemporal structures of the semidiscrete three-soliton solution determined by solution (53) are simulated in Figure 7 by selecting , , , , ,, , , , , and . In Figure 8, two catching up processes between the bell-shaped three-soliton solutions are simulated. We can see in the U-shaped trajectory that the high soliton surpasses the sub-high soliton twice and there happen two complete collisions between the high soliton and sub-high soliton. When , the corresponding three-soliton propagation in the negative direction of n-axis is shown in Figure 9. In this process of linear propagation, there is only a complete collision between the high soliton and sub-high soliton.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

5. Conclusion

In summary, we have derived and solved the new and more general semidiscrete system (1) with time-varying coefficients. To the best of our knowledge, the obtained N-soliton solutions (53) and (54) are new, and they have not been reported in the literature. When , with the help of Mathematica 4.0, the validity of the one-soliton solutions (69) and (70) has been verified by inserting them back into original system (1). Owing to the embedded finite time-varying coefficient functions, the obtained soliton solutions (53) and (54) possess rich spatiotemporal structures and more freedom to discuss the dynamical evolution of soliton solutions. In this paper, the bell-shaped one-, two-, and three-soliton solutions determined by solutions (53) and (54) are shown under the condition of . In this case, the one-, two-, and three-solitons keep soliton characteristics, and the spatiotemporal structures of and are very similar. But when we select small values of , , and , the simulations always output the disordered spatiotemporal structures, and even some of them have many singularities, which have not been shown in this paper.

It is shown in Figures 1–10 that these embedded time-varying coefficient functions , , , and affect the spatiotemporal structure and propagation velocity of semidiscrete solitons. The influences from the coefficient functions include not only the propagation speed but also the propagation direction. To some extent, this can also be verified from a mathematical point of view. For example, solutions (69) and (70) can be rewritten as

(a)

(b)

(c)

Then, the velocity of the one soliton determined by equation (77) has the following expression:

Obviously, velocity (79) depends on the coefficient functions and the constant but remains unchanged for any fixed when these coefficient functions are constants. Since the velocity affects the propagation trajectory of the one soliton, the coefficient functions affect the spatiotemporal structure of the one soliton.

Data Availability

The data used to supported the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (11547005 and 11847011), the Natural Science Foundation of Education Department of Liaoning Province of China (LZ2017002), Innovative Talents Support Program in Colleges and Universities of Liaoning Province of China (LR2016021), and the Natural Science Foundation of Liaoning Province of China (20170540007).

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Copyright © 2020 Sheng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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