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

Ma, Hongcai; Zhang, Caoyin; Deng, Aiping

doi:https://doi.org/10.1155/2020/4270906

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 4270906 | https://doi.org/10.1155/2020/4270906

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

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

Academic Editor: Toshikazu Kuniya

Received09 Mar 2020

Accepted08 Jun 2020

Published11 Jul 2020

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 isand and are defined bywhere , are all real numbers. By substituting equation (4) into equation (2), we can get the breather solution of equation (1), which isand and are defined bywhere , , 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 getand and are given bywhere , , , , , , and are some free real numbers.

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

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 byand and are defined bywhere , are all real numbers. Substituting (11) into (2), we can get the solutions of (1):and , , and are given bywhere , 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 getand , , and are determined bywhere , , , , , , , 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:

4.1.2. Case 2

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

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

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 bywhere three linear wave variables are defined bywhere , are all real numbers. Substituting (21) into (2), we can get the solutions of (1):where three linear wave variables are defined bywhere , 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 getwhere three linear wave variables are determined bywhere , , , , , , , and are free real numbers.

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

4.2.2. Case 2

where , , and are some free real numbers. Substituting (18) into (23), we getwhere three linear wave variables are given bywhere , , , , , , , and are free real numbers.

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

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 bywhere , are real numbers to be determined. Substituting (31) into (2), we can get the interaction solutions of (1):and , , and are defined bywhere , 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 getand , , and are given bywhere , , , , , , , , 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)

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 byand , , and are defined bywhere , are real numbers. Substituting (39) into (2), we can get the interaction solutions of (1):and , , and are defined bywhere , 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 getand , , and are given bywhere , , , , , , , 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)

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.

References

R. Zhang, S. Bilige, T. Fang, and T. Chaolu, “New periodic wave, cross-kink wave and the interaction phenomenon for the Jimbo-Miwa-like equation,” Computers and Mathematics with Applications, vol. 78, no. 3, pp. 754–764, 2019.
View at: Publisher Site | Google Scholar
O. Ilhan and J. Manafian, “Periodic type and periodic cross-kink wave solutions to the (2 + 1)-dimensional breaking soliton equation arising in fluid dynamics,” Modern Physics Letters B, vol. 33, no. 23, Article ID 1950277, 2019.
View at: Publisher Site | Google Scholar
W.-X. Ma, “Lump solutions to the Kadomtsev-Petviashvili equation,” Physics Letters A, vol. 379, no. 36, pp. 1975–1978, 2015.
View at: Publisher Site | Google Scholar
S. Y. Lou, “Consistent riccati expansion for integrable systems,” Studies in Applied Mathematics, vol. 134, no. 3, pp. 372–402, 2015.
View at: Publisher Site | Google Scholar
J.-Y. Yang, W.-X. Ma, and Z. Qin, “Lump and lump-soliton solutions to the (2 + 1)-dimensional Ito equation,” Analysis and Mathematical Physics, vol. 8, no. 3, pp. 427–436, 2018.
View at: Publisher Site | Google Scholar
W.-X. Ma, X. Yong, and H.-Q. Zhang, “Diversity of interaction solutions to the (2 + 1)-dimensional Ito equation,” Computers & Mathematics with Applications, vol. 75, no. 1, pp. 289–295, 2018.
View at: Publisher Site | Google Scholar
H.-Q. Zhang and W.-X. Ma, “Lump solutions to the (2 + 1)-dimensional Sawada-Kotera equation,” Nonlinear Dynamics, vol. 87, pp. 2305–2310, 2017.
View at: Publisher Site | Google Scholar
V. E. Zaharov, “Exact solutions in the problem of parametric interaction of three-dimensional wave packets,” Doklady Akademii Nauk Sssr, vol. 228, no. 6, pp. 1314–1316, 1976.
View at: Google Scholar
X. Lü, W.-X. Ma, and C. M. Khalique, “A direct bilinear Bäcklund transformation of a (2 + 1)-dimensional Korteweg-de Vries-like model,” Applied Mathematics Letters, vol. 50, pp. 37–42, 2015.
View at: Publisher Site | Google Scholar
X. Lü, J.-P. Wang, F.-H. Lin, and X.-W. Zhou, “Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water,” Nonlinear Dynamics, vol. 91, no. 2, pp. 1249–1259, 2018.
View at: Publisher Site | Google Scholar
Y. Tang, S. Tao, and Q. Guan, “Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations,” Computers & Mathematics with Applications, vol. 72, no. 9, pp. 2334–2342, 2016.
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
B. Ren, J. Lin, and Z.-M. Lou, “Consistent Riccati expansion and rational solutions of the Drinfel’d-Sokolov-Wilson equation,” Applied Mathematics Letters, vol. 105, Article ID 106326, 2020.
View at: Publisher Site | Google Scholar
B. Ren, W.-X. Ma, and J. Yu, “Characteristics and interactions of solitary and lump waves of a (2 + 1)-dimensional coupled nonlinear partial differential equation,” Nonlinear Dynamics, vol. 96, no. 1, pp. 717–727, 2019.
View at: Publisher Site | Google Scholar
B. Ren, W.-X. Ma, and J. Yu, “Rational solutions and their interaction solutions of the (2 + 1)-dimensional modified dispersive water wave equation,” Computers & Mathematics with Applications, vol. 77, no. 8, pp. 2086–2095, 2019.
View at: Publisher Site | Google Scholar
H.-C. Ma and A. P. Deng, “Lump solutions to the (2 + 1)-dimensional shallow water wave equation,” Thermal Science, vol. 21, no. 4, pp. 1765–1769, 2017.
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
H. Ma, X. Meng, H. Wu, and A. Deng, “A class of lump solutions for Ito equation,” Thermal Science, vol. 23, no. 4, pp. 2205–2210, 2019.
View at: Publisher Site | Google Scholar
H. Ma, Y. Bai, and A. Deng, “Multiple lump solutions of the (2 + 1)-dimensional Konopelchenko-Dubrovsky equation,” Mathematical Methods in the Applied Sciences, 2020, In press.
View at: Publisher Site | Google Scholar
H.-C. Ma, H. Wu, and A. Deng, “Novel interaction phenomena of localised waves in the (2 + 1)-dimensional HSI equation,” East Asian Journal on Applied Mathematics, vol. 10, no. 3, pp. 485–498, 2020.
View at: Publisher Site | Google Scholar
Y. Tang, M. Yuen, and L. Zhang, “Double wronskian solutions to the (2 + 1)-dimensional Broer-Kaup-Kupershmidt equation,” Applied Mathematics Letters, vol. 105, Article ID 11672270, 2020.
View at: Publisher Site | Google Scholar
J. Q. Lü, S. D. Bilige, and T. haolu, “The study of lump solution and interaction phenomenon to (2 + 1)-dimensional generalized fifth-order kdV equation,” Nonlinear Dynamics, vol. 91, no. 3, pp. 1669–1676, 2018.
View at: Publisher Site | Google Scholar
W. X. Ma, “Generalized bilinear differential equations,” Studies in Nonlinear Sciences, vol. 2, no. 4, pp. 140–144, 2011.
View at: Google Scholar
B. Ren and W.-X. Ma, “Rational solutions of a (2 + 1)-dimensional Sharma-Tasso-Olver equation,” Chinese Journal of Physics, vol. 60, pp. 153–157, 2019.
View at: Publisher Site | Google Scholar
Y. He, S. Li, and Y. Long, “Exact solutions to the Sharma-Tasso-Olver equation by using Improved G′/G-expansion method,” Journal of Applied Mathematics, vol. 2013, Article ID 247234, 6 pages, 2013.
View at: Publisher Site | Google Scholar
M. N. Ali, S. M. Husnine, A. Saha, S. K. Bhowmik, S. Dhawan, and T. Ak, “Exact solutions, conservation laws, bifurcation of nonlinear and supernonlinear traveling waves for Sharma-Tasso-Olver equation,” Nonlinear Dynamics, vol. 94, no. 3, pp. 1791–1801, 2018.
View at: Publisher Site | Google Scholar
Z. Yan and S. Lou, “Soliton molecules in Sharma-Tasso-Olver-Burgers equation,” Applied Mathematics Letters, vol. 104, Article ID 106271, 2020.
View at: Publisher Site | Google Scholar
S. Wang, X.-y. Tang, and S.-Y. Lou, “Soliton fission and fusion: burgers equation and Sharma-Tasso-Olver equation,” Chaos, Solitons & Fractals, vol. 21, no. 1, pp. 231–239, 2004.
View at: Publisher Site | Google Scholar
J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983.
View at: Publisher Site | Google Scholar
K. A. Gepreel, T. A. Nofal, and N. S. Al-Sayali, “Direct method for solving nonlinear strain wave equation in microstructure solids,” International Journal of Physical Sciences, vol. 11, no. 10, pp. 121–131, 2016.
View at: Publisher Site | Google Scholar
S.-f. Tian and H.-q. Zhang, “A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 173–186, 2011.
View at: Publisher Site | Google Scholar
S.-F. Tian and H.-Q. Zhang, “Riemann theta functions periodic wave solutions and rational characteristics for the (1 + 1)-dimensional and (2 + 1)-dimensional Ito equation,” Chaos, Solitons & Fractals, vol. 47, pp. 27–41, 2013.
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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