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Breather Wave Solutions and Interaction Solutions for Two Mixed CalogeroBogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko Equations.

1. Introduction

Recently, great attention has been paid to the study about exact solutions of nonlinear partial differential equations. So, it becomes more important to seek exact solutions of nonlinear partial differential equations (NLPDEs), which occur in many fields, such as chemistry, biology, optics, classical mechanics, acoustics, engineering, and social sciences. At present, many mathematicians have proposed a large number of methods to seek exact solutions, such as Backlund transformation [1], Hirota bilinear methods [2], homoclinic breather limit approach [3, 4], and Darboux transformation [5-12]. Among these methods, the Hirota bilinear method is one of the most critical and powerful methods. Recently, some new exact solutions of nonlinear partial differential equations have been constructed [13-22] by means of bilinear operator theories, so it has become an important research direction to study the dynamic properties of these new equations. In this article, the breather wave solutions will be discussed. On the basis of lump solution [23], the interaction solutions will be obtained.

The two mixed Calogero-Bogoyavlenskii-Schiff (CBS) and Bogoyavlensky-Konopelchenko (BK) equations [23] are usually written as

[mathematical expression not reproducible], (1)

where [u.sub.x] = w and [[delta].sub.i], i = 1, ..., 6, are arbitrary constants. When the constants satisfy [[delta].sub.3] = [[delta].sub.4] = [[delta].sub.5] = [[delta].sub.6] = 0, and [[delta].sub.5] = [[delta].sub.6] = 0, the Calogero-Bogoyavlenskii-Schiff (CBS) and Bogoyavlensky-Konopelchenko (CBS-BK) equations will become a generalized Calogero-Bogoyavlenskill-Schiff (gCBS) equation [24] and a generalized Calagero-Bogoyavlenskii Konopelchenko equation [25], respectively. The CBS equation was first constructed by Bogoyavlenskii and Schiff in different ways [26, 27]. Namely, Bogoyavlenskii used the modified Lax formalism, whereas Schiff derived the same equation by reducing the self-dual Yang-Mills equation. In 2019, a class explicit lump solutions of the CBS-BK equation are constructed by using the Hirota bilinear approaches by Ren et al. [23]. The (2 + 1)-dimensional CBS equation also can be derived from the Korteweg-de Vries equation [28, 29]. Moreover, the BK equation is used as the interaction of a Rieman wave propagation [30], so we called the (2 + 1)-dimensional nonlinear partial differential equation (1) as gCBS-BK equation. These two equations have been widely studied in different ways [29, 31-40].

2. The Bilinear Equation for gCBS-BK Equation

If we take

u = 2[[partial derivative].sub.x] ln f, (2)

where f(x, y, t) is an unknown real function, the bilinear equation of Equation (1) can be presented

[mathematical expression not reproducible], (3)

where [D.sub.t], [D.sub.x] are all bilinear derivative operators and D -operator [2] is defined by

[mathematical expression not reproducible], (4)

where m and t are the positive integers, a(x, t) is the function of x and t, and b(x, t) is the function of the formal variables x' and t'.

3. Breather Wave Solutions of CBSBK Equation

In this section, we will use the extended homoclinic text method [41, 42] to get the breather wave solutions of Equation (1). To start with,

f(x, y, t) = [k.sub.1] exp ([[xi].sub.1]) + exp (-[[xi].sub.1]) + [k.sub.2] cos ([[xi].sub.2]) + [a.sub.9], (5)

where [[xi].sub.1] and [[xi].sub.2] are defined by

[mathematical expression not reproducible], (6)

where [a.sub.i], i = 1, ..., 8, [k.sub.1], and [k.sub.2] are all real numbers. Substituting Equation (5) into Equation (3), we can get the following.where [a.sub.1], [a.sub.5], and [a.sub.7] are some free real numbers.

Case 1.

[mathematical expression not reproducible]. (7)

Substituting Equation (7) into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible]. (8)

where [[xi].sub.1] and [[xi].sub.2] are given by

[mathematical expression not reproducible]. (9)

where [a.sub.i], i = 1, ..., 8, [[delta].sub.1], [[delta].sub.2], [[delta].sub.5], and [[delta].sub.6] are real numbers. Figure 1 described the evolution of solution (8).

Case 2.

[mathematical expression not reproducible] (10)

Substituting Equation (10) into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible], (11)

where [[xi].sub.1] and [[xi].sub.2] are determined by

[mathematical expression not reproducible], (12)

where [a.sub.1], [a.sub.4], [a.sub.5], [a.sub.7], and [a.sub.8] are real numbers. Therefore, the dynamic behavior can be performed in Figure 2.

Case 3.

[mathematical expression not reproducible]. (13)

Substituting Equation (13) into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible], (14)

where [[xi].sub.1] and [[xi].sub.2] are given by

[mathematical expression not reproducible], (15)

where [a.sub.1], [a.sub.4], [a.sub.5], [a.sub.7], [a.sub.8], [k.sub.1], [k.sub.2], and [[delta].sub.4] are free real numbers. Figure 3 described the evolution of solution (14).

Case 4. Substituting [a.sub.1], [k.sub.1], and [a.sub.7] into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible]. (16)

The evolution of solution (16) is described in Figure 4. [[xi].sub.1] and [[xi].sub.2] are given by

[mathematical expression not reproducible], (17)

where [a.sub.2], [a.sub.3], [a.sub.4], [a.sub.5], [a.sub.6], [a.sub.8], [[delta].sub.1], and [[delta].sub.2] are real numbers.

Case 5.

[mathematical expression not reproducible]. (18)

Substituting Equation (18) into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible]. (19)

The evolution of solution (19) is described in Figure 5. [[xi].sub.1] and [[xi].sub.2] are given by

[mathematical expression not reproducible]. (20)

where [a.sub.1], [a.sub.4], [a.sub.5], [a.sub.6], [a.sub.8], [k.sub.1], [k.sub.2], [[delta].sub.3], and [[delta].sub.6] are free real numbers. The three-dimensional dynamic figure can be drawn as Figure 5.

Case 6.

[mathematical expression not reproducible], (21)

Substituting Equation (21) into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible], (22)

where [[xi].sub.1] and [[xi].sub.2] are defined by

[mathematical expression not reproducible], (23)

where [a.sub.2], [a.sub.4], [a.sub.5], [a.sub.7], [a.sub.8], [k.sub.1], [k.sub.2], [[delta].sub.1], [[delta].sub.5], and [[delta].sub.6] are some free real numbers. The figure is given as Figure 6.

Case 7.

[mathematical expression not reproducible], (24)

where [a.sub.1], [a.sub.6], [a.sub.7], [k.sub.2], [[delta].sub.1], [[delta].sub.2], [[delta].sub.3], and [[delta].sub.5] are free real numbers. Substituting Equation (24) into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible]. (25)

The figure is given as Figure 7. [[xi].sub.1] and [[xi].sub.2] are followed by

[mathematical expression not reproducible], (26)

where [a.sub.1], [a.sub.4], [a.sub.5], [a.sub.6], [a.sub.7], [a.sub.8], [k.sub.2], [[delta].sub.2], [[delta].sub.3], and [[delta].sub.5] are free real numbers.

Case 8.

[mathematical expression not reproducible], (27)

where [a.sub.1], [a.sub.6], [a.sub.7], [k.sub.2], [[delta].sub.1], [[delta].sub.2], [[delta].sub.3], and [[delta].sub.5] are free real numbers. Substituting Equation (27) into Equation (5), through the transformation (2), we have

[mathematical expression not reproducible]. (28)

The figure is drawn as Figure 8. [[xi].sub.1] and [[xi].sub.2] are defined by

[mathematical expression not reproducible], (29)

where [a.sub.1], [a.sub.2], [a.sub.4], [a.sub.6], [a.sub.7], [a.sub.8], [k.sub.2], [[delta].sub.1], [[delta].sub.2], [[delta].sub.3], and [[delta].sub.5] are free real numbers.

4. Interaction Solutions of CBS-BK System

4.1. Interaction between a Lump and One-Kink Soliton. With the help of Maple, we will discuss the interaction between a lump and one-kink soliton by taking f (x, y, t) as a combination of positive quadratic function and one exponential function, that is,

[mathematical expression not reproducible], (30)

where [[xi].sub.1], [[xi].sub.2], and [[xi].sub.3] are defined by

[mathematical expression not reproducible], (31)

where [a.sub.i], i = 1, ..., 9, [p.sub.1], [k.sub.1], [r.sub.1], and [q.sub.1] are all real numbers. In order to get the interaction solutions of Equation (1), substituting Equation (30) into Equation (2),

[mathematical expression not reproducible], (32)

where [[xi].sub.1], [[xi].sub.2], and [[xi].sub.3] are defined by

[mathematical expression not reproducible], (33)

where [a.sub.i], i = 1, ..., 9, [p.sub.1], [k.sub.1], [r.sub.1], and [q.sub.1] are all real numbers. Substituting Equation (30) into Equation (3), through complex analysis and calculations, we can have the following.

Case 1.

[mathematical expression not reproducible], (34)

where [a.sub.2], [a.sub.3], [a.sub.5], [a.sub.7], [p.sub.1], and [[delta].sub.6] are free real numbers. Substituting Equation (34) into Equation (32), we have

[mathematical expression not reproducible]. (35)

Case 2.

[mathematical expression not reproducible], (36)

where [a.sub.2], [a.sub.6], [a.sub.7], [k.sub.1], [p.sub.1], [[delta].sub.3], and [[delta].sub.4] are some free real numbers. Substituting Equation (36) into Equation (32), we have

[mathematical expression not reproducible], (37)

where [[xi].sub.1], [[xi].sub.2], and [[xi].sub.3] are defined by

[mathematical expression not reproducible], (38)

where [a.sub.2], [a.sub.4], [a.sub.6], [a.sub.7], [a.sub.8], [k.sub.1], [p.sub.1], [r.sub.1], [[delta].sub.3], and [[delta].sub.4] are some free real numbers.

In order to obtain the dynamic feature, we choose Case 2 to analyse. The three-dimensional dynamic graphs are drawn as Figure 9. We can find that the lump waves and the exponential function waves interact with each other and keep moving in the opposite direction.

4.2. Interaction between a Lump and Periodic Waves. In order to get interaction solutions between a lump and periodic waves, we will take f as the combination of positive function and hyperbolic cosine function. Therefore, f can be determined by

[mathematical expression not reproducible], (39)

where variables are defined by

[mathematical expression not reproducible], (40)

where [a.sub.i], i = 1, ..., 9, [p.sub.1], [k.sub.1], [r.sub.1], and [q.sub.1] are all real numbers. Substituting Equation (39) into Equation (2), we can get the interaction solutions of Equation (1):

[mathematical expression not reproducible], (41)

where [[xi].sub.1], [[xi].sub.2], and [[xi].sub.3] are defined by

[mathematical expression not reproducible], (42)

where [a.sub.i], i = 1, ..., 9, [p.sub.1], [k.sub.1], [r.sub.1], and [q.sub.1] are real numbers. Through long and tedious calculations, we can get the following relations between the parameters.where [a.sub.5], [a.sub.7], [b.sub.1], [p.sub.1], [[delta].sub.1], [[delta].sub.2], and [[delta].sub.6] are free real numbers.where [a.sub.5], [a.sub.6], [a.sub.7], [p.sub.1], [b.sub.1], [[delta].sub.1], and [[delta].sub.6] are free real numbers.where [a.sub.2], [a.sub.3], [a.sub.5], [a.sub.7], [p.sub.1], and [[delta].sub.6] are some free real numbers.where [a.sub.5], [p.sub.1], [[delta].sub.1], and [[delta].sub.6] are free real numbers.

Case 1.

[mathematical expression not reproducible], (43)

Case 2.

[mathematical expression not reproducible], (44)

Case 3.

[mathematical expression not reproducible], (45)

Case 4.

[mathematical expression not reproducible], (46)

When we change the coefficients of the equation, the value of Equation (47) will be different accordingly. In order to obtain the dynamic feature, we choose Case 2 to analyse. Taking Equation (44) into Equation (41), we can get

[mathematical expression not reproducible]. (47)

With the help of Maple, the three-dimensional dynamic graphs are drawn as Figure 10. We can find that lump waves and periodic waves interact with each other and keep moving in the opposite direction.

5. Conclusions

In this paper, based on a bilinear differential equation, we study the breather wave solutions and the interaction solutions of the mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko equations. Compared with the existing results in the literature, our results are new. It will be extensively used to report many attractive physical phenomena in the fields of acoustics, heat transfer, fluid dynamics, classical mechanics, and so on. It is demonstrated that the Hirota operators are very simple and powerful in constructing new nonlinear differential equations, which possess nice math properties. It is interesting to study the interaction solutions between soliton solutions and period solution by making f as a combination of exponential function and trigonometry function. However, this method can be applied to those equations which have Hirota bilinear forms. Furthermore, we also can study the quardrilinear forms and even polylinearity forms of this equation in the future. These questions may also be interesting and worth studying.

https://doi.org/10.1155/2020/1458280

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 is 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 Nonlinear Sciences, and the Fundamental Research Funds for the Central Universities.

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Hongcai Ma [ID], (1,2) Caoyin Zhang, (1) and Aiping Deng (1,2)

(1) Department of Applied Mathematics, Donghua University, Shanghai 201620, China

(2) Institute for Nonlinear Sciences, Donghua University, Shanghai 201620, China

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

Received 10 March 2020; Revised 12 May 2020; Accepted 20 May 2020; Published 16 June 2020

Academic Editor: Antonio Scarfone

Caption: FIGURE 1: Spatiotemporal structure of solution (8) with the parameter selections [a.sub.1] = 1, [a.sub.2] = 1, [a.sub.3] = 1, [a.sub.6] = 1, [k.sub.1] = 1, [[delta].sub.1] = 1, [[delta].sub.2] = 1, [[delta].sub.5] = 1, and [[delta].sub.6] = 1.

Caption: FIGURE 2: Spatiotemporal structure of solution (11) with the parameter selections [a.sub.1] = 1, [a.sub.5] = 1, and [a.sub.7] = 1.

Caption: FIGURE 3: Spatiotemporal structure of solution (14) with the parameter selections [a.sub.1] = 1, [a.sub.5] = 1, [a.sub.7] = 1, [k.sub.1] = 1, [k.sub.2] = 1, and [[delta].sub.4] = 1.

Caption: FIGURE 4: Spatiotemporal structure of solution (16) with the parameter selections [a.sub.2] = 1, [a.sub.5] = 1, [a.sub.6] = 1, [k.sub.2] = 1, [[delta].sub.1] = 1, [[delta].sub.2] = 1, [[delta].sub.3] = 1, and [[delta].sub.5] = 1.

Caption: FIGURE 5: Spatiotemporal structure of solution (19) [a.sub.1] = 1, [a.sub.5] = 1, [a.sub.6] = 1, [k.sub.1] = 1, [k.sub.2] = 1, [[delta].sub.3] = 1, and [[delta].sub.6] = 1.

Caption: FIGURE 6: Spatiotemporal structure of solution (22) with the parameter selections [a.sub.2] = 1, [a.sub.5] = 1, [a.sub.7] = 1, [k.sub.1] = 1, [k.sub.2] = 1, [[delta].sub.1] = 1, [[delta].sub.5] = 1, and [[delta].sub.6] = 1.

Caption: Figure 7: Spatiotemporal structure of solution (25) with the parameter selections [a.sub.1] = 1, [a.sub.6] = 1, [a.sub.7] = 1, [k.sub.2] = 1, [[delta].sub.1] = 1, [[delta].sub.2] = 1, [[delta].sub.3] = 1, and [[delta].sub.5] = 1.

Caption: FIGURE 8: Spatiotemporal structure of solution (28) with the parameter selections [a.sub.1] = 1, [a.sub.6] = 1, [a.sub.7] = 1, [k.sub.2] = 1, [[delta].sub.1] = 1, [[delta].sub.2] = 1, [[delta].sub.3] = 1, and [[delta].sub.5] = 1.

Caption: Figure 9: Spatiotemporal structure of solution (37) with the parameter selections: (a) t = -10, [a.sub.2] = 1, [p.sub.1] = 1, [a.sub.3] = 1, [a.sub.7] = 1, [a.sub.5] = 1, and [[delta].sub.6] = 1; (b) t = 0, [a.sub.2] = 1, [p.sub.1] = 1, [a.sub.3] = 1, [a.sub.7] = 1, [a.sub.5] = 1, and [[delta].sub.6] = 1; (c) t = 10, [a.sub.2] = 1, [p.sub.1] = 1, [a.sub.3] = 1, [a.sub.7] = 1, [a.sub.5] = 1, and [[delta].sub.6] = 1.

Caption: FIGURE 10: Spatiotemporal structure of solution (47) with the parameter selections: (a) t = 0, [a.sub.5] = 1, [a.sub.6] = 1, [a.sub.7] = 1, p, = 1, b, = 1, [[delta].sub.1] = 1, and [[delta].sub.6] = 1; (b) t = 10, [a.sub.5] = 1, [a.sub.6] = 1, [a.sub.7] = 1, [p.sub.1] = 1, [b.sub.1] = 1, [[delta].sub.1] = 1, and [[delta].sub.6] = 1; (c) [a.sub.5] = 1, [a.sub.6] = 1, [a.sub.7] = 1, [p.sub.1] = 1, [b.sub.1] = 1, [[delta].sub.1] = 1, and [[delta].sub.6] = 1.
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Title Annotation:Research Article
Author:Ma, Hongcai; Zhang, Caoyin; Deng, Aiping
Publication:Advances in Mathematical Physics
Date:Jun 30, 2020
Words:4071
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