The homoclinic breather wave solution, rational wave and n-soliton solution to a nonlinear differential equation
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Zhenzhen Zheng
and
Tao Xu
Abstract
According to the homoclinic breather limit method, we obtain the homoclinic breather wave and rational wave of a nonlinear evolution differential equation. The n-soliton wave solutions are derived by utilizing the Hirota method. In addition, the graphs of these solutions are shown by selecting the appropriate parameters.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11871232,11501526
Funding source: Natural Science Foundation of Henan Province
Award Identifier / Grant number: 212300410417
Acknowledgement
This work is supported by the National Natural Science Foundation of China (grant number: 11871232) and the Natural Science Foundation of Henan Province (grant number: 212300410417) and the Training Plan of Young Key Teachers in Universities of Henan Province.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: None declared.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge, Cambridge University Press, 1991.10.1017/CBO9780511623998Search in Google Scholar
[2] M. Maeda, H. Sasaki, and E. Segawa, “Scattering and inverse scattering for nonlinear quantum walks,” Discrete Contin. Dyn. Syst., vol. 38, pp. 3687–3703, 2018. https://doi.org/10.3934/dcds.2018159.Search in Google Scholar
[3] S. Randoux, P. Suret, and G. El, “Inverse scattering transform analysis of rogue waves using local periodization procedure,” Sci. Rep., vol. 6, pp. 29238–29251, 2016. https://doi.org/10.1038/srep29238.Search in Google Scholar PubMed PubMed Central
[4] H. Q. Zhao, J. Y. Yuan, and Z. N. Zhu, “Integrable semi-discrete Kundu-Eckhaus equation: Darboux transformation, breather, rogue wave and continuous limit theory,” J. Nonlinear Sci., vol. 28, pp. 43–68, 2018. https://doi.org/10.1007/s00332-017-9399-9.Search in Google Scholar
[5] V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Berlin, Springer, 1991.10.1007/978-3-662-00922-2Search in Google Scholar
[6] Q. Zhao and L. H. Wu, “Darboux transformation and explicit solutions to the generalized TD equation,” Appl. Math. Lett., vol. 67, pp. 1–6, 2017. https://doi.org/10.1016/j.aml.2016.11.012.Search in Google Scholar
[7] B. Q. Li and Y. L. Ma, “Lax pair, Darboux transformation and N th-order rogue wave solutions for a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation,” Comput. Math. Appl., vol. 77, pp. 514–524, 2019. https://doi.org/10.1016/j.camwa.2018.09.054.Search in Google Scholar
[8] M. J. Dong, S. F. Tian, X. W. Yan, and L. Zou, “The (3+1)-dimensional Hirota bilinear equation; bilinear form; solitary waves; rogue waves; homoclinic breather waves,” Comput. Math. Appl., vol. 75, pp. 957–964, 2018. https://doi.org/10.1016/j.camwa.2017.10.037.Search in Google Scholar
[9] R. Hirota, The Direct Method in Soliton, Cambridge, Cambridge University Press, 2004.10.1017/CBO9780511543043Search in Google Scholar
[10] H. Gao, “Dynamics of Nth-order rogue waves in (2+1)-dimensional Hirota equation,” Pramana - J. Phys., vol. 88, pp. 84–88, 2017. https://doi.org/10.1007/s12043-017-1392-1.Search in Google Scholar
[11] H. Gao, “Rogue waves, homoclinic breather waves and soliton waves for the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation,” Appl. Math. Lett., vol. 65, pp. 90–97, 2017.10.1016/j.aml.2016.10.009Search in Google Scholar
[12] R. R. Jia and R. Guo, “Breather and rogue wave solutions for the (2+1)-dimensional nonlinear Schrodinger–Maxwell–Bloch equation,” Appl. Math. Lett., vol. 93, pp. 117–123, 2019. https://doi.org/10.1016/j.aml.2019.02.001.Search in Google Scholar
[13] Z. D. Dai, C. J. Wang, and J. Liu, “Inclined periodic homoclinic breather and rogue waves for the (1+1)-dimensional Boussinesq equation,” Pramana - J. Phys., vol. 83, pp. 473–480, 2014. https://doi.org/10.1007/s12043-014-0811-9.Search in Google Scholar
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- Limit cycles in a tritrophic food chain model with general functional responses
- Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations
- Stress concentration effect on deflection and stress fields of a master leaf spring through domain decomposition and geometry updation technique
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