Complexity
Volume 2020 (2020), Article ID 4270906, 8 pages
https://doi.org/10.1155/2020/4270906

Research Article

New Periodic Wave, Cross-Kink Wave, Breather, and the Interaction Phenomenon for the (2 + 1)-Dimensional Sharmo–Tasso–Olver Equation

Hongcai Ma,1,2 Caoyin Zhang,¹ and Aiping Deng1,2

¹Department of Applied Mathematics, Donghua University, Shanghai 201620, China
²Institute for Nonlinear Sciences, Donghua University, Shanghai 201620, China

Correspondence should be addressed to Hongcai Ma; hongcaima@hotmail.com

Received 9 March 2020; Accepted 8 June 2020; Published 11 July 2020

Academic Editor: Toshikazu Kuniya

Copyright © 2020 Hongcai Ma 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.

Abstract.

In this paper, with the aid of symbolic computation, several kinds of exact solutions including periodic waves, cross-kink waves, and breather are proposed by using a trilinear form for the (2 + 1)-dimensional Sharmo–Tasso–Olver equation. Then, by combing the different forms, the interactions between a lump and one-kink soliton and between a lump and periodic waves are generated. Moreover, the dynamic characteristics of interaction solutions are analyzed graphically by selecting suitable parameters with the help of Maple.

1. Introduction

In soliton theory, the study of exact solutions of nonlinear partial differential equations (NLPDES) has attracted more and more attention. Therefore, finding exact solutions of nonlinear partial differential equations is becoming more and more important for the study of oceanographic engineering, atmosphere, chemistry, biology, finance, and social science. With the aid of Maple or Mathematica, one can get the complex solutions including periodic wave, cross-kink wave, and breather of nonlinear partial differential equations. For example, the complex solutions of the Jimbo-Miwa-like equation [1] and the (2 + 1)-dimensional breaking soliton equation [2] have been obtained. Later on, interaction solutions to lump-kink, lump-soliton, and lump periodic waves of many NLPDES have been presented so that the complex solutions can be constructed. To get the interaction between the lump and other nonlinear waves, Ma, Lou, and Yang et al. obtained some new ways by using the bilinear method and CRE method [3–5], which have been applied in many fields [6–23]. All of them are very important and useful. In this article, based on a trilinear form, our main purpose is to study the periodic waves, cross-kink waves, breather, and interaction solutions between lump-soliton and lump periodic waves of the (2 + 1)-dimensional Sharmo–Tasso–Olver equation by combining a positive quadratic function with an exponential function or a trigonometric function. The (2 + 1)-dimensional Sharmo–Tasso–Olver equation is usually written as follows [24]:

where is an analytic function.

The abovementioned equation is a generalization of the (1 + 1)-dimensional Sharmo–Tasso–Olver equation [25–27]. It can describe the propagation of a nonlinear dispersive wave of inhomogeneous media. The generalized Kaup–Newell-type hierarchy of nonlinear evolution equations is explicitly related to the Sharma-Tasso-Olver equation. By the Hirota direct method and the Bäcklund transformation, the fission and fusion of the solitary waves have been obtained. [28].

In this paper, based on a transformation, we mainly introduce a trilinear form of generalized the (2 + 1)-dimensional Sharmo–Tasso–Olver equation. In Section 3, we present a breather of the (2 + 1)-dimensional Sharmo–Tasso–Olver equation. In Section 4, we study periodic waves and cross-kink waves of the (2 + 1)-dimensional Sharmo–Tasso–Olver equation. In Section 5, with the help of analysis and symbolic computations, by mixing a positive quadratic function with an exponential function or a trigonometric function, the interaction between a lump and one-kink soliton and the interaction between a lump and periodic waves of equation (1) are studied. At the same time, plots are presented to show the change of the equation, and the interactional phenomena are discussed.

2. The Trilinear Equation for the Sharmo–Tasso–Olver Equation

Hirota bilinear forms play an important role in solving the lump solutions. With the widespread use of the bilinear form, trilinear forms and even polylinearity forms gradually appear. By truncated Painleve analysis [29], we can get the solutions of (1) as follows:

where is an unknown real function. Through equation (2), the trilinear equation of (1) can be presented as follows:

where and are the arbitrary positive numbers. Next, we will use equation (3) to get a series of solutions.

3. Breather Wave Solutions for the Sharmo–Tasso–Olver Equation

We use the extended homoclinic test method [30–32] to construct as a form of solutions, which is

and and are defined by

where , are all real numbers. By substituting equation (4) into equation (2), we can get the breather solution of equation (1), which is

and and are defined by

where , , are all real numbers. By substituting (4) into (3), we can get some relations of parameters as follows:

where , , and are some free real numbers.

Substituting (8) into (6), we can get

and and are given by

where , , , , , , and are some free real numbers.

Therefore, the change of the equation is described in Figure 1.

Figure 1: Spatiotemporal structure of solution (9) with parameter selections , , and .

4. Diversity of Wave Solutions

4.1. The Periodic Cross-Kink Wave Solutions of the (2 + 1)-Dimensional Sharmo–Tasso–Olver Equation

In order to study the periodic cross-kink waves of the (2 + 1)-dimensional Sharmo–Tasso–Olver equation, we assume that the solutions for (1) are determined by

and and are defined by

where , are all real numbers. Substituting (11) into (2), we can get the solutions of (1):

and , , and are given by

where , are all real numbers. Substituting (11) into (3), we can get some relations between parameters.

4.1.1. Case 1

where , , , , and are free real numbers. Taking (15) into (3), we can get

and , , and are determined by

where , , , , , , , and are free real numbers.

The value of (16) will change when coefficients of the equation are replaced by suitable values. The three-dimensional dynamic Figure 2 is plotted as follows:

Figure 2: Spatiotemporal structure of solution (16) with parameter selections: , , , , and .

4.1.2. Case 2

where and are free real numbers. Substituting (18) into (13), we can get

and , , and are given by

where , , , , , , , , and are free real numbers.

So, we can draw Figure 3 to describe change of the equation.

Figure 3: Spatiotemporal structure of solution (19) with parameter selections: , , and .

4.2. The Periodic Wave Solutions of the Sharmo–Tasso–Olver Equation

For purpose of getting the periodic wave solutions of the (2 + 1)-dimensional Sharmo–Tasso–Olver equation, we assume that the solutions for (1) are determined by

where three linear wave variables are defined by

where , are all real numbers. Substituting (21) into (2), we can get the solutions of (1):

where three linear wave variables are defined by

where , are real numbers. Substituting (21) into (3), we can get the relations of parameters as follows:

4.2.1. Case 1

where , , , , and are free real numbers. Substituting (25) into (23), we can get

where three linear wave variables are determined by

where , , , , , , , and are free real numbers.

So, we can draw Figure 4 to describe change of the equation.

Figure 4: Spatiotemporal structure of solution (23) with parameter selections: , , , , and .

4.2.2. Case 2

where , , and are some free real numbers. Substituting (18) into (23), we get

where three linear wave variables are given by

where , , , , , , , and are free real numbers.

So, we can draw Figure 5 to describe change of the equation.

Figure 5: Spatiotemporal structure of solution (28) with the parameter selections: , , and .

5. Interaction Solutions of the Sharmo–Tasso–Olver Equation

5.1. Interaction between a Lump Wave and One-Kink Soliton

In this section, in order to get the interaction between a lump wave and one-kink soliton of (1), we assume as a combination of a positive quadratic function and an exponential function:

and , , and are defined by

where , are real numbers to be determined. Substituting (31) into (2), we can get the interaction solutions of (1):

and , , and are defined by

where , are real numbers. Substituting (31) into (3), we can get relations between the parameters as follows.

5.1.1. Case 1

where , , , , , and are free real numbers.

5.1.2. Case 2

where , , , and are free real numbers.

In order to analyze the dynamics properties concisely, we choose case 1 to analyze. Substituting (35) into (33), we can get

and , , and are given by

where , , , , , , , , and are real numbers.

The three-dimensional dynamic figures of the waves are shown in Figure 6. We can find that the lump waves and the exponential function waves will interact with each other and keep moving forward.

(a)

(b)

(c)

Figure 6: Spatiotemporal structure of solution (37) with parameter selections: : , , , , , , ; : , , , , , , ; : , , , , , , and .

5.2. Interaction between a Lump Wave and Periodic Waves

In the previous section, we have obtained interaction solutions between a lump and one-kink soliton of the (2 + 1)-dimensional Sharmo–Tasso–Olver equation. In this part, we will discuss the interaction between a lump wave and periodic waves by combining a positive function with a hyperbolic cosine function. We assume that the solutions for (1) is determined by

and , , and are defined by

where , are real numbers. Substituting (39) into (2), we can get the interaction solutions of (1):

and , , and are defined by

where , are real numbers. Substituting (39) into (3), we can obtain the following relations between parameters.

5.2.1. Case 1

where , , , , , , and are some free real numbers.

5.2.2. Case 2

where and are some free real numbers.

In order to analyze the dynamic properties concisely, we choose case 1 to analyze. Sustituting (43) into (41), we can get

and , , and are given by

where , , , , , , , and are real numbers.

The three-dimensional dynamic figures of the waves are shown in Figure 7. We can find that the lump waves and the periodic waves will interact with each other and keep moving forward.

(a)

(b)

(c)

Figure 7: Spatiotemporal structure of solution (45) with the parameter selections: (a) , , , , , , ; (b) , , , , , , ; (c) , , , , , , and .

6. Conclusions

In this paper, we derived the periodic wave, cross-kink wave, breather, and the interaction solutions, such as the interaction between a lump and one-kink soliton and the interaction between a lump and periodic waves of the (2 + 1)-dimension Sharmo–Tasso–Olver equation. By using the trilinear form, some interaction solutions of the (2 + 1)-dimension Sharmo–Tasso–Olver equation have been obtained with symbolic computations. The graphs of their interaction evolution processes over time are presented, and their dynamic characteristics are analysed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (project nos. 11371086, 11671258, and 11975145), the Fund of Science and Technology Commission of Shanghai Municipality (project no. 13ZR1400100), the Fund of Donghua University, institute for non-linear sciences, and the Fundamental Research Funds for the Central Universities.

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