Breather Wave Solutions and Interaction Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko Equations

Ma, Hongcai; Zhang, Caoyin; Deng, Aiping

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

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2020 | Article ID 1458280 | https://doi.org/10.1155/2020/1458280

Breather Wave Solutions and Interaction Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko Equations

Hongcai Ma,^1,2Caoyin Zhang,¹and Aiping Deng^1,2

Academic Editor: Antonio Scarfone

Received10 Mar 2020

Revised12 May 2020

Accepted20 May 2020

Published16 Jun 2020

Abstract

In this paper, based on a bilinear differential equation, we study the breather wave solutions by employing the extended homoclinic test method. By constructing the different forms, we also consider the interaction solutions. Furthermore, it is natural to analyse dynamic behaviors of three-dimensional plots.

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 Bäcklund 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 where and , , are arbitrary constants. When the constants satisfy , and , 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 ()-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 ()-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 where is an unknown real function, the bilinear equation of Equation (1) can be presented where , are all bilinear derivative operators and -operator [2] is defined by where and are the positive integers, is the function of and , and is the function of the formal variables and .

3. Breather Wave Solutions of CBS-BK 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, where and are defined by where , , , and are all real numbers. Substituting Equation (5) into Equation (3), we can get the following.where , , and are some free real numbers.

Case 1. Substituting Equation (7) into Equation (5), through the transformation (2), we have where and are given by where , , , , , and are real numbers. Figure 1 described the evolution of solution (8).

Case 2.

Substituting Equation (10) into Equation (5), through the transformation (2), we have where and are determined by where , , , , and are real numbers. Therefore, the dynamic behavior can be performed in Figure 2.

Case 3.

Substituting Equation (13) into Equation (5), through the transformation (2), we have where and are given by where , , , , , , , and are free real numbers. Figure 3 described the evolution of solution (14).

Case 4. Substituting , , and into Equation (5), through the transformation (2), we have

The evolution of solution (16) is described in Figure 4.

and are given by where , , , , , , , and are real numbers.

Case 5.

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

The evolution of solution (19) is described in Figure 5.

and are given by where , , , , , , , , and are free real numbers. The three-dimensional dynamic figure can be drawn as Figure 5.

Case 6.

Substituting Equation (21) into Equation (5), through the transformation (2), we have where and are defined by where , , , , , , , , , and are some free real numbers. The figure is given as Figure 6.

Case 7. where , , , , , , , and are free real numbers. Substituting Equation (24) into Equation (5), through the transformation (2), we have

The figure is given as Figure 7.

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

Case 8. where , , , , , , , and are free real numbers. Substituting Equation (27) into Equation (5), through the transformation (2), we have

The figure is drawn as Figure 8.

and are defined by where , , , , , , , , , , and 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 as a combination of positive quadratic function and one exponential function, that is, where , , and are defined by where , , , , , and are all real numbers. In order to get the interaction solutions of Equation (1), substituting Equation (30) into Equation (2), where , , and are defined by where , , , , , and are all real numbers. Substituting Equation (30) into Equation (3), through complex analysis and calculations, we can have the following.

Case 1. where , , , , , and are free real numbers. Substituting Equation (34) into Equation (32), we have

Case 2. where , , , , , , and are some free real numbers. Substituting Equation (36) into Equation (32), we have where , , and are defined by where , , , , , , , , , and 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.

(a)

(b)

(c)

4.2. Interaction between a Lump and Periodic Waves

In order to get interaction solutions between a lump and periodic waves, we will take as the combination of positive function and hyperbolic cosine function. Therefore, can be determined by where variables are defined by where , , , , , and are all real numbers. Substituting Equation (39) into Equation (2), we can get the interaction solutions of Equation (1): where , , and are defined by where , , , , , and are real numbers. Through long and tedious calculations, we can get the following relations between the parameters.where , , , , , , and are free real numbers.where , , , , , , and are free real numbers.where , , , , , and are some free real numbers.where , , , and are free real numbers.

Case 1.

Case 2.

Case 3.

Case 4.

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

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.

(a)

(b)

(c)

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 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.

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.

References

C. Rogers and W. F. Shadwick, Backlund Transformations and Their Applications, Academic Press, New York, 1982.
R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, United Kingdom, 2004.
C. Y. Qin, S. F. Tian, X. B. Wang, and T. T. Zhang, “On breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 62, pp. 378–385, 2018.
View at: Publisher Site | Google Scholar
X.-w. Yan, S.-f. Tian, M.-j. Dong, L. Zhou, and T.-t. Zhang, “Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation,” Computers & Mathematics with Applications, vol. 76, no. 1, pp. 179–186, 2018.
View at: Publisher Site | Google Scholar
X. N. Gao, S. Y. Lou, and X. Y. Tang, “Bosonization, singularity analysis, nonlocal symmetry reductions and exact solutions of supersymmetric KdV equation,” Journal of High Energy Physics, vol. 2013, no. 5, p. 29, 2013.
View at: Publisher Site | Google Scholar
B. Ren, “The (2+1)-dimensional Konopelchenko–Dubrovsky equation: nonlocal symmetries and interaction solutions,” Nonlinear Dynamics, vol. 86, no. 3, pp. 1855–1862, 2016.
View at: Publisher Site | Google Scholar
B. Ren, “Symmetry reduction related with nonlocal symmetry for Gardner equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 42, pp. 456–463, 2017.
View at: Publisher Site | Google Scholar
Y. L. Dang, H. J. Li, and J. Lin, “Soliton solutions in nonlocal nonlinear coupler,” Nonlinear Dynamics, vol. 88, no. 1, pp. 489–501, 2017.
View at: Publisher Site | Google Scholar
J. Lin, X. W. Jin, X. L. Gao, and S. Y. Lou, “Solitons on a periodic wave background of the modified KdV-Sine-Gordon equation,” Communications in Theoretical Physics, vol. 70, no. 2, pp. 119–126, 2018.
View at: Publisher Site | Google Scholar
Y. Yıldırım and E. Yaşar, “Multiple exp-function method for soliton solutions of nonlinear evolution equations,” Chinese Physics B, vol. 26, no. 7, p. 070201, 2017.
View at: Publisher Site | Google Scholar
A. R. Adem, Y. Yildirim, and E. Yaşar, “Soliton solutions to the non-local Boussinesq equation by multiple exp-function scheme and extended Kudryashov’s approach,” Pramana - Journal of Physics, vol. 92, no. 2, pp. 1–11, 2019.
View at: Google Scholar
Y. Yildäräm, E. Yasar, A. R. Adem, and E. Yasar, “A novel scheme for nonlinear evolution equations using symbolic computations,” Journal of Applied Nonlinear Dynamics, vol. 8, no. 3, pp. 463–473, 2019.
View at: Publisher Site | Google Scholar
X. Lü, S.-T. Chen, and W.-X. Ma, “Constructing lump solutions to a generalized Kadomtsev–Petviashvili–Boussinesq equation,” Nonlinear Dynamics, vol. 86, no. 1, pp. 523–534, 2016.
View at: Publisher Site | Google Scholar
W.-x. Ma, Y. Zhang, Y. Tang, and T. Junyi, “Hirota bilinear equations with linear subspaces of solutions,” Applied Mathematics and Computation, vol. 218, no. 13, pp. 7174–7183, 2012.
View at: Publisher Site | Google Scholar
C.-G. Shi, B.-Z. Zhao, and W.-X. Ma, “Exact rational solutions to a Boussinesq-like equation in (1+1)-dimensions,” Applied Mathematics Letters, vol. 48, pp. 170–176, 2015.
View at: Publisher Site | Google Scholar
Y. Zhang and W.-X. Ma, “Rational solutions to a KdV-like equation,” Applied Mathematics and Computation, vol. 256, pp. 252–256, 2015.
View at: Publisher Site | Google Scholar
H.-C. Ma and A.-P. Deng, “Lump solution of (2+1)-dimensional Boussinesq equation,” Communications in Theoretical Physics, vol. 65, no. 5, pp. 546–552, 2016.
View at: Publisher Site | Google Scholar
Y. Yıldırım and E. Yaşar, “An extended Korteweg–de Vries equation: multisoliton solutions and conservation laws,” Nonlinear Dynamics, vol. 90, no. 3, pp. 1571–1579, 2017.
View at: Publisher Site | Google Scholar
E. Yaşar, Y. Yıldırım, and A. R. Adem, “Extended transformed rational function method to nonlinear evolution equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 20, no. 6, pp. 691–701, 2019.
View at: Publisher Site | Google Scholar
Y. Yıldırım and E. Yasar, “Wronskian solutions of (2+1)-dimensional nonlocal Ito equation,” Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, vol. 67, no. 2, pp. 126–138, 2018.
View at: Publisher Site | Google Scholar
A. R. Adem, Y. Yildirim, and E. Yaşar, “Complexiton solutions and soliton solutions: (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation,” Pramana, vol. 92, no. 3, p. 36, 2019.
View at: Publisher Site | Google Scholar
Y. Yildirim, E. Yasar, and A. R. Adem, “A multiple exp-function method for the three model equations of shallow water waves,” Nonlinear Dynamics, vol. 89, no. 3, pp. 2291–2297, 2017.
View at: Publisher Site | Google Scholar
B. Ren, W. X. Ma, and J. Yu, “Lump solutions for two mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko equations,” Communications in Theoretical Physics, vol. 71, no. 6, pp. 658–662, 2019.
View at: Publisher Site | Google Scholar
S.-T. Chen and W.-X. Ma, “Lump solutions of a generalized Calogero–Bogoyavlenskii–Schiff equation,” Computers & Mathematics with Applications, vol. 76, no. 7, pp. 1680–1685, 2018.
View at: Publisher Site | Google Scholar
S.-T. Chen and W.-X. Ma, “Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation,” Frontiers of Mathematics in China, vol. 13, no. 3, pp. 525–534, 2018.
View at: Publisher Site | Google Scholar
O. I. Bogoyavlenskii, “Overturning solitons in new two-dimensional integrable equations,” Mathematics of the USSR-Izvestiya, vol. 34, no. 2, pp. 245–259, 1990.
View at: Publisher Site | Google Scholar
J. Schiff, “Integrability of Chern-Simons-Higgs vortex equations and a reduction of the self-dual Yang-Mills equations to three dimensions,” in Painleve Trascendents,Their Asymptotics and Physical Applications, D. Levi, Ed., pp. 393–405, Plenum Press, New York, 1992.
View at: Google Scholar
K. Toda and S.-j. Yu, “The investigation into the Schwarz–Korteweg–de Vries equation and the Schwarz derivative in (2+1) dimensions,” Journal of Mathematical Physics, vol. 41, no. 7, pp. 4747–4751, 2000.
View at: Publisher Site | Google Scholar
M. S. Bruzon, M. L. Gandarias, C. Muriel, J. Ramirez, S. Saez, and F. R. Romero, “The Calogero-Bogoyavlenskii-Schiff equation in 2+1 dimensions,” Theoretical and Mathematical Physics, vol. 137, no. 1, pp. 1367–1377, 2003.
View at: Publisher Site | Google Scholar
B. G. Konopelchenko, Solitons in Multidimensions: Inverse Spectrum Transform Method, 1993.
F. Calogero, “A method to generate solvable nonlinear evolution equations,” Lettere Al Nuovo Cimento, vol. 14, no. 12, pp. 443–447, 1975.
View at: Publisher Site | Google Scholar
S.-J. Yu, K. Toda, N. Sasa, and T. Fukuyama, “Nsoliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3 1) dimensions,” Journal of Physics A: Mathematical and General, vol. 31, no. 14, pp. 3337–3347, 1998.
View at: Publisher Site | Google Scholar
K. Toda, S.-j. Yu, and T. Fukuyama, “The Bogoyavlenskii-Schiff hierarchy and integrable equations in (2 + 1) dimensions,” Reports on Mathematical Physics, vol. 44, no. 1-2, pp. 247–254, 1999.
View at: Publisher Site | Google Scholar
A.-m. Wazwaz, “The (2+1) and (3+1)-dimensional CBS equations: multiple soliton solutions and multiple singular soliton solutions,” Zeitschrift für Naturforschung A, vol. 65, no. 3, pp. 173–181, 2010.
View at: Publisher Site | Google Scholar
Taogetusang, “New infinite sequence soliton-like solutions of (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation,” Acta Physica Sinica, vol. 62, no. 21, p. 210201, 2013.
View at: Google Scholar
B. Li and Y. Chen, “Exact analytical solutions of the generalized Calogero-Bogoyavlenskii-Schiff equation using symbolic computation,” Czechoslovak journal of physics, vol. 54, no. 5, pp. 517–528, 2004.
View at: Publisher Site | Google Scholar
H. Yan, S. F. Tian, L. L. Feng, and T. T. Zhang, “Quasi-periodic wave solutions, soliton solutions, and integrability to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation,” Waves in Random and Complex Media, vol. 26, no. 4, pp. 444–457, 2016.
View at: Publisher Site | Google Scholar
H. Triki, Z. Jovanoski, and A. Biswas, “Shock wave solutions to the Bogoyavlensky-Konopelchenko equation,” Indian Journal of Physics, vol. 88, no. 1, pp. 71–74, 2014.
View at: Publisher Site | Google Scholar
Y.-L. Wang, Y.-T. Gao, S.-L. Jia, G.-F. Deng, and H. Wen-Qiang, “Solitons for a (2+1)-dimensional variable-coefficient Bogoyavlensky-Konopelchenko equation in a fluid,” Modern Physics Letters B, vol. 31, no. 25, p. 1750216, 2017.
View at: Publisher Site | Google Scholar
J. Manafian, B. M. Ivatloo, and M. Abapour, “Breather wave, periodic, and cross-kink solutions to the generalized Bogoyavlensky-Konopelchenko equation,” Mathematical Methods in the Applied Sciences, vol. 43, no. 4, pp. 1753–1774, 2019.
View at: Google Scholar
Z. Dai, J. Liu, and D. Li, “Applications of HTA and EHTA to YTSF Equation,” Applied Mathematics and Computation, vol. 207, no. 2, pp. 360–364, 2009.
View at: Publisher Site | Google Scholar
Z. Xu, H. Chen, and Z. Dai, “Rogue wave for the (2+1)-dimensional Kadomtsev–Petviashvili equation,” Applied Mathematics Letters, vol. 37, pp. 34–38, 2014.
View at: Publisher Site | Google Scholar

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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.

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