Abstract

In this work, we investigate a linear partial differential equation in (3+1)-dimensions. We construct lump solutions which localize in all directions in the -space. By combining the trigonometric, hyperbolic and exponential functions with a quadratic function, diversity interaction solutions such as interacted lumps with periodic waves and interacted lumps with multisoliton are generated. The phenomena of interaction solutions between a lump and a multisoliton and between a lump and a multikink soliton are presented by figures. The results expand understanding dynamical behavior of the (3+1)-dimensional partial differential equations.

1. Introduction

Nonlinear partial differential equations (NLPDEs) have attracted an increasingly attention from mathematics and physicists. Finding exact solutions to NLPDEs is one of the fundamental topics in this field [1–4]. Among these exact solutions, lump solutions, another kind of rational solutions, was first discovered by Manakov et al. [5]. Lump waves are found to be localized in all directions of the space. The subject of lump waves has a widespread application in the field of mathematical physics and engineering [6–9].

Due to an important physical significance of lump waves, many high-dimensional nonlinear partial differential equations admit lump solutions, such as the Ishimori-I equation [10], the Kadomtsev-Petviashvili (KP) equation [11], the extended KP equation [12], the KP-Boussinesq equation [13, 14], the Hirota bilinear equation [15], the generalized Hirota-Satsuma-Ito equation [16], and the generalized KP equation [17]. Furthermore, interaction solutions among solitons and other kinds of complicated waves are studied by the localization procedure related to the nonlocal symmetry [18–20]. Similar to these interaction solutions, interactive lump-kink [21–24] and lump-soliton [25–27] of the nonlinear evolution equations have been studied by combining a positive quadratic function with an exponential function. A special rogue wave of the (2+1)-dimensional Korteweg-de Vries equation [28] and the KP equation [29] is obtained by combining the positive quadratic function and the hyperbolic cosine function. In the (3+1)-dimensional case, only lump-type solutions which are rationally localized in almost all but not all directions in space [25–27, 30–33] can be presented. In addition, the lump, interaction between a lump and periodic wave, and interaction between a lump and one soliton of a (3+1)-dimensional linear partial differential equation (PDE) are explored by symbolic computations [34]. It will be interesting to enlarge this category of the (3+1)-dimensional PDE that possesses lump solutions. In this paper, the main purpose of this work is to construct a (3+1)-dimensional linear PDE which possesses lump solutions. In the meanwhile, interaction solutions between lumps and multisoliton are obtained by introducing an ansätz function. This kinds of interaction solutions to the (3+1)-dimensional PDE have not been reported in other studies.

This paper is organized as follows. In Section 2, we study a class of (3+1)-dimensional linear PDE. The lumps, interaction between lumps and periodic waves, and interaction solutions between lumps and multisoliton are derived by using an ansätz function. Some concluding remarks will be given in the last section.

2. Lumps and Interaction Solutions of a (3+1)-Dimensional PDE

We consider a (3+1)-dimensional linear PDE:where are arbitrary constants.

A kind of exact solutions assumeswhere is an arbitrary real function and are four linear variables:where , , , , and are real constants to be determined. By substituting (2) into (1), the linear PDE (1) becomeswhere and are quadratic and one-time functions of the parameters , , , , and . By setting and for all present combinations of and , we can get the following equations on the parameters:By solving the above determined equations, we can obtain several solutions of the constraint constants. We just list one case of the constraint constants:

The explicit exact solutions can be expressed asorwithwhere , , and are arbitrary natural numbers, are defined byand the function is an arbitrary function of . We can takewhere , , , , , and are constants to guarantee the positivity of . By selecting the function , we can obtain lump waves, interaction between a lump and periodic waves, interaction solution between a lump and one soliton, interaction solution between a lump and a one-kink soliton, and interaction solution between a lump and a two-stripe soliton to the linear PDE (1). To describe the interaction solution between a lump and one soliton, we set the parameters asThe 3D plot and density plot for the interaction solution between a lump and one soliton of (9) are shown in Figure 1. The 3D plot and density plot for the interaction solution between a lump and a one-kink soliton of (10) are depicted in Figure 2.

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Figure 1 
The interaction solution between a lump and one soliton of (9). (a) 3-dimensional plot with the time ; (b) the corresponding density plot.
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Figure 2 
The interaction solution between a lump and a one-kink soliton of (10). (a) 3-dimensional plot with the time ; (b) the corresponding density plot.

To get another kind of exact solutions, we assumewhere is an arbitrary real function, and are five linear variables:By substituting (15) into (1), the linear PDE (1) becomeswhere and are quadratic and one-time functions of the parameters , , , , and . By the accurate calculation, we can get several constraining equations on the various parameters. We list one case of the constraint parameters:

The explicit exact solutions can be expressed asorwithwhere , , and are arbitrary natural numbers, and the functions and are arbitrary functions of and , respectively. The lumps, interaction between lumps and periodic waves, and interaction between lumps and solitons can be obtained by selecting arbitrary functions and . We can takewhere , , , , , and are constants to guarantee the positivity of . To describe the interaction solutions between a lump and two solitons and between a lump and a two-kink soliton, we set the parameters asThe 3D plot for the interaction solution between a lump and two solitons of (19) is depicted in Figure 3(a). The interaction solution between a lump and a two-kink soliton of (20) is shown in Figure 3(b).

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Figure 3 
(a) 3-dimensional plot of the interaction solution between a lump and two solitons (19) with the time ; (b) 3-dimensional plot of the interaction solution between a lump and a two-kink soliton (20) with the time .

We can assume another kind of solutions aswhere is an arbitrary real function, and are six linear variables:Based on the same process, we can get the following set of the constraining parameters:The explicit exact solutions can be expressed asorwithwhere , , and are arbitrary natural numbers, and functions , , and are arbitrary functions of , , and , respectively. We can takewhere , , , , , and are constants to guarantee the positivity of . The lumps, interaction between lumps and periodic waves, and interaction between lumps and solitons can be derived by selecting arbitrary functions , , and . To describe interaction solution between a lump and three solitons, we set the parameters asThe graphical behavior of the interaction solution between a lump and three solitons of (27) is shown in Figure 4(a). The graphical behavior of the interaction solution between a lump and a three-kink soliton of (28) is analyzed in Figure 4(b). To the best of our knowledge, these interaction solutions of the (3+1)-dimensional PDE have not been reported in other studies.

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Figure 4 
(a) 3-dimensional plot of the interaction solution between a lump and three solitons of (27) with the time ; (b) 3-dimensional plot of the interaction solution between a lump and a three-kink soliton of (28) with the time .

Remark. From the constraining parameters of (8), (18), and (26), a determinant equation satisfiesIt is shown that the rational solutions of (9), (10), (19), (20), (27), and (28) fail to be localized in all directions in the whole -space. The rational solutions of (9), (10), (19), (20), (27), or (28) are all lump waves, rationally localized in all directions in the -space [34].

3. Conclusion

In summary, some novel interaction solutions of a (3 + 1)-dimensional PDE are considered by using the ansätz functions (2), (15), and (24). By combining a quadratic function with exponential, hyperbolic, or trigonometric functions, lumps, interaction between lumps and periodic waves and interaction solutions between lumps and multisoliton are found. The phenomena of interaction solution between a lump and a multisoliton for the (3+1)-dimensional PDE are generated as illustrative examples.

In this paper, the behavior of interaction between a lump and three solitons has been displayed for the choices of specific parameters. We can get interaction solutions between lumps and multisoliton by using the following ansätz function:where is an arbitrary real function, and are multilinear variables. The results obtained in this paper provide further evidence on the effectiveness of lump waves in (3+1)-dimensions. Besides, some exact solutions to the (3+1)-dimensional nonlinear evolution equation can be constructed by use of the test function method [35]. The study of those problems may be helpful in soliton theory.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China no. 11775146.