Abstract
In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, the N-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.
1. Introduction
With recent advancements in nonlinear science, the application of nonlinear evolution equations (NLEEs) has gained more prominence amongst mathematicians and physicists. The use of NLEEs is important because NLEEs can be used to describe many interesting nonlinear dynamic behaviors in nonlinear fields like optical fibers, plasma physics, atmospheric science, biologic nerve propagation, and marine science. Although the use of NLEEs is significant, the task of finding an explicit solution for the NLEEs has always been difficult and tedious. However, this task is also an important and meaningful exercise.
Many effective methods for solving NLEEs have been proposed by previous researchers [1–9]. For instance, Zhang et al. obtained the exact solutions for generalized nonlinear diffusion equations by using the separation of variables method [4]. He and Wu also used the Exp-function method to solve the nonlinear wave equations [5]. Xu et al. presented analytic solutions of the discrete PT-symmetric nonlocal nonlinear Schrödinger equation by Darboux transformation [9]. Amongst the proposed methods [1–9], the Hirota bilinear method is one of the most direct and effective methods for obtaining the exact solution of the NLEEs.
The most critical step in using the Hirota bilinear method is to transform the original nonlinear equation into a bilinear equation by introducing appropriate variable transformations. After obtaining the bilinear equation, the multisoliton solution of the original nonlinear equation can then be constructed. Interestingly, Ma et al. showed that multivariable polynomials can also produce Hirota bilinear equations [10]. The necessary criterion for the existence of a linear subspace for the exponential traveling wave solutions of the Hirota bilinear equation was given by Ma et al. [10]. Generally, the Hirota bilinear equations derived from multivariable polynomials have linear subspace solutions.
By considering the multivariable polynomial given by [10]where are arbitrary constants whose corresponding Hirota bilinear equation iswhere and is the Hirota bilinear operator [2].
If the coefficients are chosen as and , then (2) can be reduced to the bilinear form of the Korteweg–de Vries (KdV) equation. Furthermore, if the coefficients are set as , , and , then (2) can be reduced to the bilinear form of (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation.
By introducing the variable transformation given by(2) is then mapped towhere and . Based on our literature review, (5) is a generalized (3 + 1)-dimensional NLEE which has not been presented by previous authors. Recently, many generalized NLEEs have attracted more attention as a result of their widespread application in many physical fields [11–14]. By investigating different kinds of generalized NLEEs, more general properties of the equations can be obtained.
This article is divided into sections and it focuses on the generalized (3 + 1)-dimensional NLEE represented by equation (5). In Section 2, the bilinear Bäcklund transformation (BBT) of equation (5) is constructed based on its Hirota bilinear equation. Armed with the BBT, the traveling wave solution and a mixed solution of (5) are obtained. In Section 3, the one-soliton, the two-soliton, and the three-soliton solutions are obtained, and then the N-soliton solution is derived. In Section 4, the lump and the interaction solutions of (5) are given. The solutions also included some free parameters based on the bilinear form of (5). However, their dynamical behaviors are shown under some selected values of parameters. The article is then concluded with a summary and the implications of this research.
2. Bilinear Bäcklund Transformation and Its Application
Bäcklund transformation is important in the study of soliton systems. Some bilinear identities are given as follows [2]:where and .
2.1. Bilinear Bäcklund Transformation
The BBT for (2) is then constructed and summarized as follows.
Theorem 1. If satisfies (2) as well as the following relations:then also satisfies (2) with (11)–(14), where are arbitrary constants.
Proof. For the following function,we show that satisfying (11)–(14) can make .Based on (9) and (10) respectively, it is clear that the coefficients of and are zeros. Hence, the proof of Theorem 1 is complete.
2.2. Application of BBT
The value is the solution of (2) and the corresponding original variable is . Due to , BBT (11)–(14) associated with is reduced to the following equations:
Different types of solutions to (5) can be obtained by substituting different forms of into (17).
2.2.1. Traveling Wave Solution
First consider the following form:where , , , , and are undetermined constants. By inserting (18) into (17) and assuming , these parameters can be determined by calculation as follows:
Therefore,
This solves (5), where , , , , and are arbitrary constants and . Solution (20) is a one-stripe soliton solution of (5).
By taking a simple polynomial solution,where , , , and are constants to be determined. Similarly, by substituting (21) into (17) and supposing , a direct computation getswith and , , and are arbitrary constants.
Hence,is a rational solution to (5), and parameters , , , and satisfy (22).
2.2.2. Mixed Solution
is now taken as a combination of a polynomial function and cosine function:where , , , , and are constants to be determined. By inserting (24) into (17), these parameters can be determined by calculation as follows:where , , , , and are arbitrary constants and and .
Thus,is a mixed form solution to (5), and , , , and satisfy (25).
3. Soliton Solutions
It is known that many interesting solutions, including breathers, lump, peakon, and rogue solutions, are all special reductions of N-soliton solutions in different situations. This section shows solutions with a few solitons to (5). The standard perturbation method is used to find possible soliton-like solutions. By using a tedious but straightforward calculation, the one-soliton, two-soliton, three-soliton, and N-soliton solutions can be obtained.
3.1. One-Soliton Solution
To obtain the one-soliton solution, we assume that in (2) is expressed aswithwhere , , , , and are arbitrary constants. By substituting (27) into (2) and making the coefficients of all exponential terms zero, the dispersion relation is obtained as follows:
Therefore, by substituting (27)–(29) into (4), the one-soliton solution of (5) can be obtained.
3.2. Two-Soliton Solution
To obtain the two-soliton solution, let in (2) be expressed aswithwhere , , , , and are arbitrary constants. By substituting (30) into (2) and making the coefficients of all exponential terms zero and after some reduction, the followings are obtained:
By substituting (30)–(32) into (4), the two-soliton solution of (5) can be obtained.
3.3. Three-Soliton Solution
To obtain the three-soliton solution, it is assumed that in (2) is expressed aswithwhere , , , , and are arbitrary constants. Similar to the above method, but through a more complex process and then reduction, the following results are finally obtained:
By substituting (33)–(35) into (4), the three-soliton solution of (5) can be obtained. When the three-soliton are substituted into (4) for verification, it is shown that (35) needs to satisfy the constraint condition .
Figure 1 shows the three-soliton solution of (5) through selecting the appropriate parameters. As it is shown in Figure 1, the wave shapes are kink solitons. Both one and two solitons are also kink solitons, whose graphs are omitted here.
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3.4. N-Soliton Solution
(5) admits multiple-soliton solution (4) with of the following form:wherewith , , , and being arbitrary constants, , and is the summation that over all possible combinations of . By substituting (36) and (37) into (4), the N-soliton solution of (5) can be obtained.
4. Lump and Interaction Solutions
4.1. Lump Solutions
Many studies of the literature have given positive quadratic functions to get lump solutions [15–17]. Based on these studies of the literature, we have the following:where , and are real parameters to be determined.
By substituting into (2) along with a direct but tedious calculation, 12 sets of solutions are obtained. By removing the redundant parameter solutions, the following three cases are finally obtained.
Case 1. where , , and , are arbitrary constants.
By substituting (38) and (39) into (4), the lump solution of (5) is as follows:where and satisfy (39).
Case 2. where , , and , are arbitrary constants.
By substituting (38) and (41) into (4), the lump solution of (5) is as follows:where satisfy (41). The first set parameter solution in (41) is then chosen. Figure 2 shows the spatial structures of the lump solution and the contour plots for (42) by selecting the appropriate parameter values.
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Case 3. wherewhere , , and , , , and are arbitrary constants.
By substituting (38) and (43) into (4), the lump solution of (5) is obtained as follows:where satisfy (43).
It is noteworthy to state that the spatial structures of the lump solutions in Case 1 and Case 3 are like those in Case 2. From the above three cases, the obtained lump waves have the following asymptotic behavior: , , , and .
4.2. Interaction Solution between Solitary and Lump Waves
To construct interaction solutions, the following form is taken:withwhere , , , , , , and are arbitrary parameters and is an arbitrary function. By choosing different functions for , abundant interaction solutions can be obtained [18–21].
This section mainly considers the interaction between the lump and the soliton solutions and then discusses their corresponding dynamical behaviors. The following values are chosen for and in (46) to obtain
By substituting (48) into (2) and combining direct but tedious calculations, 31 sets of solutions are obtained for the above undetermined parameters. To ensure the generality of parametric solutions, the following four cases are obtained.
Case 4. where , , and , , , and are arbitrary constants.
By substituting (48) and (49) into (4), the interaction solution of (5) is obtained as follows:where satisfy (49) and the first set parameter solution in (49) is chosen. Figure 3 shows the spatial structures of the interaction solution and the contour plots for (50) by selecting the appropriate parameter values. Interestingly, it was found that Figure 3(a) is consistent with the result of the interaction between the kink soliton and the lump waves (see Figures 1 and 2).
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Case 5. where , and , , , and are arbitrary constants.
By substituting (48) and (51) into (4), the interaction solution of (5) is as follows:where satisfy (51). Figure 4 shows the spatial structures of the interaction solution and the contour plots for (52) by selecting the appropriate parameter values.
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Case 6. where , , and , , and are arbitrary constants.
By substituting (48) and (53) into (4), the interaction solution of (5) is obtained as follows:where satisfy (53). The first set parameter solution in (53) is then chosen. Figure 5 shows the spatial structures of the interaction solution and the contour plots for (54) by selecting the appropriate parameter values.
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Case 7. where , , and , , , and are arbitrary constants.
By substituting (48) and (55) into (4), the interaction solution of (5) is obtained as follows:where satisfies (55). The first set parameter solution in (55) is then chosen. Figure 6 shows the spatial structures of the interaction solution and the contour plots for (56) by selecting the appropriate parameter values.
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5. Conclusion
In this article, a generalized (3 + 1)-dimensional NLEE whose bilinear equation is obtained by a multivariable polynomial was discussed. Based on the bilinear equation, the BBT is given and it consists of four bilinear equations involving seven arbitrary parameters except for the parameters of the equation itself. By using the BBT, two traveling wave solutions and a mixed solution were obtained. Also, the N-soliton solutions were constructed and the dynamic properties of the soliton solutions were shown graphically. Furthermore, the lump and the interaction solutions were constructed. The properties of the explicit analytic solutions were shown by using figures. The general expression of the interaction solution was also given. By determining the different forms of the arbitrary function, more abundant interaction solutions will be obtained. Hence, the interaction between the lump and other solutions, including periodic solution and kink solution, can further be explored.
Finally, this article not only presents a new example for (3 + 1)-dimensional NLEE but also provides a reference for finding the lump and the interaction solutions of other generalized multidimensional NLEEs.
Data Availability
The Maple data used to support 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.
Acknowledgments
This work was completed with the support of the National Natural Science Foundation of China (12001480 and 12071418), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2019L0868), the Applied Basic Research Program of Shanxi Province (20210302123082 and 201901D211461), and the Doctoral Research Project of Yuncheng University (YQ-2020019). The authors would like to express their gratitude to EditSprings (https://www.editsprings.com/) for the expert linguistic services provided.