Abstract
In this paper, 1-lump solution and 2-lump solution of a (2 + 1)-dimensional Sawada-Kotera-like equation are obtained by means of the Hirota’s bilinear method and long wave limit method. The propagation orbits, velocities and the collisions among waves are analyzed. By setting the parameter values, the dynamic characteristics of the obtained solutions are shown in 3D and density plots. These conclusions enrich the dynamical theory of higher-dimensional nonlinear dispersive wave equations.
1 Introduction
Nonlinear evolution equations can be used to simulate various nonlinear phenomena in the real world, which appear in fluid mechanics [1–3], optical fibers[4], applied mathematics[5–7], chemistry and biology[8–10], etc. In recent years, searching for exact solutions of nonlinear evolution equations has attracted considerable attention, such as lump solutions[11–16], soliton solutions[17–21] and breather solutions[22–25].
The (2 + 1)-dimensional Sawada–Kotera equation:has important and wide applications in conformal field theory, quantum gravity field theory and conserved current of Liouville equation[26–28]. Soliton solutions[29–31], lump solutions[32,33], travelling wave solutions[34] and some other exact solutions[35] of Eq. 1 have been detailed. In this paper, we mainly consider the (2 + 1)-dimensional Sawada-Kotera-like equation[36]:in which ∂−1 represents the partial integration operator. Eq. 2 is gained from Eq. 1 by the generalized bilinear method[36]. When vx = uy and ωx = u, Eq. 2 can be reduced to Eq. 1. And Eq. 2 is different from the Sawada-Kotera-like equations which have been mentioned by [32,37].As far as we known, multiple lump solutions of Eq. 2 have not been presented in any existing articles. Classic lump, generalized lump solutions and new rogue wave solutions of Eq. 2 have been obtained by [36]. In this paper, we will study multiple lump solutions of Eq. 2. In Section 2, we construct 1-lump solution and 2-lump solution of Eq. 2 by employing the Hirota’s bilinear method and long wave limit method. The dynamical behaviors of the solutions are analyzed in Section 3. Section 4 is our conclusions.
2 1-lump solution and 2-lump solution
The long wave limit method is an effective method to generate M-lump solutions from N-soliton solutions[38–44]. In this section, we will construct the 1-lump solution and 2-lump solution of Eq. 2. As a preparation for constructing 1-lump solution and 2-lump solution of Eq. 2, we first study the N-soliton solutions[45]. With the aid of the variable transformationEq. 2 can be transformed into a bilinear form[36]:where Dt, Dx, and Dy are the bilinear derivative operators, which can be defined by generalized D operator[46]:It means that Eq. 3 are solutions of Eq. 2 if and only if f is a solution of Eq. 4. Based on the Hirota’s bilinear method, the N-soliton solutions of Eq. 4 have been obtained[45]:wherewith ki, pi and are arbitrary constants,is the summation of possible combinations of μi = 0, 1(i = 1, 2⋯N).By taking a limit of and considering all the ki in the same asymptotic order in Eq. 6, we havewhereand represents the summation over all possible compositions of i, j, ⋯p, q, which are taken different values from 1, 2, ⋯N. Taking N = 2M in Eq. 8 and in Eq. 9, where “*” denotes the complex conjugation, the M-lump solutions to Eq. 4 can be obtained.In the case of N = 2, Eq. 8 is changed intowhere , , . If we take , where a and b are real constants, I is an imaginary number unit, Eq. 10 is changed intothen we can obtain the 1-lump solution of Eq. 2:This wave keeps moving on the line , the velocity along the x-axis is vx = x + 5b2t + 5a2t, and the velocity along the y-axis is vy = y − 10 at.In the case of N = 4, Eq. 8 is changed intowhere , and . Substituting Eq. 13 into Eq. 3, we can obtain 2-lump solution of Eq. 2:wherein which . The wave keeps moving along the line , .
Figures 1–2, show the evolution of the 1-lump solution Eq. 12 and 2-lump solution Eq. 14 with the time variation. Figure 1 show the 1-lump waves for Eq. 2 under a = 1, b = 1 but with the different values of (a) and (d) t = −1, (b) and (e) t = 0, (c) and (f) t = 1. Figure 2 are the 2-lump waves for Eq. 2 with parameters a1 = 1, , b1 = 1, with the different values of (a) and (d) t = −1, (b) and (e) t = 0, (c) and (f) t = 1. Figures 1D,E,F are the density plot of Figures 1A,B,C separately and Figures 2D,E,F are the density plot of Figures 2A,B,C respectively.
FIGURE 1
FIGURE 2
3 Conclusion
In this paper, we have presented the 1-lump solution Eq. 12 and 2-lump solution Eq. 14 of the (2 + 1)-dimensional Sawada-Kotera-like Eq. 2 by using a variable transformation. Dynamical features and density distributions of the presented solutions have been depicted through plots. It is expected that these results can be useful to understand the dynamical behavior of relevant fields in physics.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
FQ: work out the whole idea of this paper, including method and writing. SL: some calculations and writing of the paper. ZL: polish the whole paper. PW: check the English gramma.
Funding
The study is supported by the key project of Beijing Social Science Foundation “strategic research on improving the service quality of capital logistics based on big data technology (18GLA009)”.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
multiple lump solution, long wave limit, sawada-kotera-like equation, hirota bilinear, partial differential equations
Citation
Qi F-H, Li S, Li Z and Wang P (2022) Multiple lump solutions of the (2+1)-dimensional sawada-kotera-like equation. Front. Phys. 10:1041100. doi: 10.3389/fphy.2022.1041100
Received
10 September 2022
Accepted
15 September 2022
Published
06 October 2022
Volume
10 - 2022
Edited by
Yunqing Yang, Zhejiang Ocean University, China
Reviewed by
Xing Lue, Beijing Jiaotong University, China
Xiazhi Hao, Zhejiang University of Technology, China
Updates
Copyright
© 2022 Qi, Li, Li and Wang.
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*Correspondence: Shuang Li, liangsh222@gmail.com
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.