Abstract
The multilinear variable separation (MLVS) approach has been proven to be very useful in solving (2 + 1)-dimensional integrable systems. Taking the (3 + 1)-dimensional Burgers hierarchy as an example, we extend the MLVS approach to a whole family of (3 + 1)-dimensional Burgers hierarchy. New exact solutions and universal formulas are obtained, which lead to abundant (3 + 1)-dimensional coherent structures. In particular, two ring-type soliton molecules and their interactions are shown in detail. We also generalize the MLVS results of the (3 + 1)-dimensional Jimbo–Miwa (JM) equation and modified JM equation.
1. Introduction
Recently, soliton theory has been widely applied in almost all branches of physics, like condensed matter physics, quantum field theory, fluid dynamics, and nonlinear optics. One of the most important research fields in soliton theory is to construct exact solutions of nonlinear evolution equations (NLEEs), which can be used to simulate natural phenomena. It is well known that the Fourier transform and the variable separation approach are the two most effective ways to find exact solutions for linear equations. The inverse scattering transformation serves as a nonlinear Fourier transform for integrable NLEEs. However, it is very difficult to extend the variable separation approach to nonlinear cases effectively. Fortunately, the multilinear variable separation (MLVS) approach has been proposed and proved to be a powerful method for finding exact solutions of many (2 + 1)-dimensional NLEEs [1–12] like the Davey–Stewartson equation, the Nizhnik–Novikov–Veselov equation, the Broer–Kaup–Kupershmidt equation and the long wave-short wave interaction equation. We call all these NLEEs the MLVS solvable models. We also can see that a (2 + 1)-dimensional universal formula is valid for suitable fields or potentials of MLVS solvable NLEEs, where at least one low-dimensional arbitrary function can be included. Thus, a large class of (2 + 1)-dimensional coherent structures, such as dromions, lumps, ring-solitons, breathers, instantons, and compactons, have been obtained. These studies are restricted to single-valued situations. For more complicated cases, multivalued functions also have been used to construct folded solitary waves and foldons. In [2, 3], it is also pointed out that the interactions among ring-solitons and some types of compactons are completely elastic, and the interactions among peakons or among some other types of compactons are not completely elastic because their shapes are changed during the interactions.
However, the MLVS approach has rarely been successfully generalized to higher dimensional NLEEs and hierarchies. As far as we know, there are only the (3 + 1)-dimensional Burgers equation [4, 5] and Jimbo–Miwa (JM) equation [8] that have been successfully solved by using this approach. Thus, our goal is primarily to apply the MLVS approach to a whole family of (3 + 1)-dimensional NLEEs, namely, the following (3 + 1)-dimensional Burgers hierarchy,where is a positive integer and are real constant parameters. This is a direct generalization of the (2 + 1)-dimensional Burgers hierarchy [12]. For , , and , Equation (1) corresponds to the (2 + 1)-dimensional Burgers equation and authors have obtained many new types of soliton structures of periodic waves [13]. For and , Equation (1) degenerates to the (1 + 1)-dimensional Burgers hierarchy [14], which corresponds to the Burgers equation or the Sharma–Tasso–Olver equation when the positive integer or .
The organization of this letter is as follows: Section 2 is devoted to describing the process of getting MLVS solutions of Equation (1) via the MLVS approach. New coherent structures, such as the two ring-type soliton molecules for the universal formula, are obtained in Section 3. A brief summary and discussions are given in Section 4.
2. MLVS Solutions of the Burgers Hierarchy
Similar to the usual steps in solving (2 + 1)-dimensional NLEEs via the MLVS approach [1–12], we construct the following auto-Bäcklund transformation first,through the truncated Laurent expansions. Here, is a suitable seed solution of Equation (1) and the unknown function needs to be determined.
It is obvious that a simple seed solution of Equation (1) can be selected as . The substitution of Equation (2) with this seed solution into Equation (1) transforms Equation (1) into the following:
To solve this equation, we change it to the form as follows:
Thus, we have the following:
Next, the second step in the MLVS approach is to make an appropriate assumption for the expansion function . Essentially, the arbitrary functions in the MLVS solutions originate from the seed solution with respect to the same variables. Since the seed solution and include three arbitrary functions of , the unknown function is taken as the following MLVS ansatz:
Substituting this expansion, Equation (6) into Equation (5) arrives at the following:
Obviously, Equation (7) can be divided into variable-separated equations, which are in the form of the following:where are arbitrary functions. In order to construct the explicit expression of the MLVS solution, we have the following:by taking without losing generality.
Because are arbitrary functions, we may treat this problem alternatively. We now consider are arbitrary functions while can be determined by Equations (9)–(11). Namely, by using Cramer’s rule in linear algebra, we can solve out and as follows:from Equations (9)– (11). In addition, for linear Equations (12)–(14), we have exact solutions of such as
Thus, exact MLVS solutions of the (3 + 1)-dimensional Burgers hierarchy (Equation (1)) are obtained.
Remark 1. The (3 + 1)-dimensional JM equation [8, 15] in its potential form, which can be expressed as follows:By means of the following Bäcklund transformation:and the MLVS ansatzwe have the following:Thus, this equation can be separated into the following:where are arbitrary functions. Because is an arbitrary function, we may treat Equation (23) alternatively. Now, we consider to be an arbitrary function while can be determined by Equation (23), that isConsidering the arbitrariness of the functions , and , it is quite straightforward to verify that the Riccati Equation (24) has an exact solution as follows:here, we rewrite , and as followsThus, the MLVS solution, Equation (20) with Equations (21), (25), and (26) of Equation (19) is obtained.
Remark 2. The (3 + 1)-dimensional modified JM (mJM) equation in its potential form, which can be expressed as follows:By using the Bäcklund transformation, Equation (20), and the MLVS ansatz, Equation (21), we have the following:So we have similar results.
3. Two Ring-Type Soliton Molecules
For (2 + 1)-dimensional MLVS solvable NLEEs, by setting the following MLVS ansatz:a quite universal formula,is derived to describe some special solutions for suitable physical quantities. In this formula, parameters , , , and are arbitrary constants, and is an arbitrary function, and may be an arbitrary function or an arbitrary solution of a special equation such as Riccati-type equation. Setting , we have . Thus, can be viewed as a basic MLVS ansatz based on the corresponding Bäcklund transformation. For the (3 + 1)-dimensional JM Equation (19) and mJM Equation (28), we have extended the universal formula (Equation (31)) to the following:where is an arbitrary function and is Equation (26). Further, for the (3 + 1)-dimensional Burger hierarchy (Equation (1)), we have two more generalized formulas as follows:
Because some arbitrary lower-dimensional functions have been included in Equations (32)–(34), we believe that abundant (3 + 1)-dimensional coherent structures can be constructed directly.
For (3 + 1)-dimensional case, it should be pointed out that we just can draw projective figures along some particular directions. So here we still give some (2 + 1)-dimensional coherent structures and their interactions. Of course, this also applies to the (3 + 1)-dimensional case, such as in Equation (32), because . We will not discuss all the possible coherent structures but only list a new particular example, the ring-type soliton molecule. There are some types of elliptic, parabolic, and hyperbolic coherent structures, which are not equal to zero identically at some quadratic curves and decays exponentially apart from the curves [2]. By restricting the functions , as some summation forms of the exponential functions as follows:and , we can obtain a structure of two ring-type solitons for Equation (32) (see Figure 1). The interaction of two ring-type solitons has been studied in [2]. If functions are changed to the following:the ring-type soliton on the right degenerates into a lump-type structure (see Figure 2). This is different from the commonly defined lump structure, which is formed by rational functions.
Next, we investigate a new coherent structure, the ring-type soliton molecule, and its interaction. It is well-known that a soliton molecule is a bound state of solitons. The research on soliton molecules has attracted considerable attention and has focused on experimental and numerical fields. Recently, many novel forms of soliton molecules have been found in (1 + 1)-dimensional and (2 + 1)-dimensional integrable systems, such as the kink-type soliton molecules, dromion molecules, and breather molecules. Here, by means of the following:and , we can obtain four ring-type solitons. Because of the special construction of the function in Equation (37), which is independent of the time variable , we have actually obtained a structure of two ring-type soliton molecules for Equation (32) (see Figure 3). In density Figures 4–8, we plot the interaction property of the two ring-type soliton molecules for Equation (32) with the selection at the times , respectively. We can see that after the head-on collision of two ring-type soliton molecules, they preserve their shapes totally. In other words, the collision between the two ring-type soliton molecules is completely elastic.
4. Summary and Discussions
Exact solutions of NLEEs play a key role in understanding possible behaviors of natural phenomena [16–18]. There are many powerful methods for constructing exact solutions of NLEEs, such as the inverse scattering transform method, the Darboux transformation, Hirota’s bilinear method, the Lie group method, and so on. The MLSA approach is one of the most efficient methods because the constructed exact solution contains low-dimensional arbitrary functions, which allows us to construct abundant coherent structures. Since the MLVS approach has been widely applied to the (2 + 1)-dimensional NLEEs [1–12], how can it work for (3 + 1)-dimensional NLEEs or hierarchies? The (3 + 1)-dimensional Burgers equation and JM equation [8, 15] are the only two NLEEs that have been solved out their MLVS solutions via this approach. In this letter, we have considered the (3 + 1)-dimensional Burgers hierarchy (Equation (1)) by using the MLVS approach. Namely, starting from the auto-Bäcklund transformation (Equation (2)) and taking MLVS ansatz (Equation (5)), we can obtain much more general exact solutions of Equation (1). In Remarks 1 and 2, we improve the results of the (3 + 1)-dimensional JM equation and construct MLVS solutions of the (3 + 1)-dimensional mJM equation, respectively. We also have investigated a new coherent structure, the ring-type soliton molecule, and its interaction.
It is worth considering to construct other forms of variable separation solutions based on the Darboux transformation. We hope that the MLVS approach can be used to solve other high-dimensional equations or hierarchies [19–21]. How to construct (3 + 1)-dimensional coherent structures and study the properties of elastic or inelastic collisions is worthy of further study.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Ethical Approval
The authors state that this research complies with the ethical standards. This research does not involve either human participants or animals.
Consent
All authors have given consent to participate in the study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed to the writing—original draft, methodology, software, formal analysis, and funding acquisition. All authors read and approved the final version of the manuscript for publication.
Acknowledgments
The authors are indebted to the referees for their meticulous reading of the manuscript and for suggestions that improved the overall presentation of the paper. The work was supported by the National Natural Science Foundation of China, no. 12375006.