Abstract
In this work, we investigate the Lakshmanan–Porsezian–Daniel (LPD) equation with high-order nonlinear and dispersion terms which can describe an inhomogeneous one-dimensional anisotropic Heisenberg ferromagnetic spin chain and alpha helical protein. With the aid of auxiliary function, the bilinear form of the LPD equation is constructed. Multi-soliton solutions are obtained by solving the corresponding bilinear form. Multi-breather solutions are presented by assuming the complex conjugation relations on the parameters of the multi-solitons. The dynamics of one breather, two breathers and their interactions are constructed by selecting the appropriate parameters. The strength of the high-order nonlinear and dispersion effects plays a key role in the breather solutions. Soliton molecule, the breather–soliton molecule and the breather molecule are discovered by applying the velocity resonance conditions. The interactions among the soliton molecules, which could be observed in marine and oceanic waters, are investigated through numerical simulation.
Similar content being viewed by others
Data availability
The data that support the findings of this study are available upon reasonable request.
References
Y.S. Kivshar, G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003)
A. Hasegawa, Optical Solitons in Fibers (Springer, 2013)
M. Daniel, K. Deepamala, Physica A 22, 241 (1995)
X.W. Jin, S.J. Shen, Z.Y. Yang, J. Lin, Phys. Rev. E 105, 014205 (2022)
R.R. Wang, Y.Y. Wang, C.Q. Dai, Opt. Laser Technol. 152, 108103 (2022)
M. Lakshmanan, Phys. Lett. A 61, 53–54 (1997)
C.J. Thompson, K.A. Ross, B.J.P. Thompson et al., Physica A 133, 330–336 (1985)
K. Nakamura, Y. Nakahara, A.R. Bishop, Phys. Rev. Lett. 54, 861 (1985)
M. Lakshmanan, K. Porsezian, M. Daniel, Phys. Lett. A 133, 483488 (1988)
K. Porsezian, M. Daniel, M. Lakshmanan, J. Math. Phys. 33, 1807–1816 (1992)
Y.F. Wang, N. Liu, B.L. Guo, J. Math. Anal. Appl. 506, 125560 (2022)
M. Wang, Y. Chen, Nonlinear Dyn. 111, 655 (2023)
R.X. Liu, B. Tian, L.C. Liu et al., Physica B 413, 120 (2013)
Y. Ye, C. Hou, D. Cheng et al., Phys. Lett. A 384, 126226 (2020)
M. Stratmann, T. Pagel, F. Mitschke, Phys. Rev. Lett. 95, 143902 (2005)
B. Orta, A. Zaviyalov, C.K. Nielsen et al., Opt. Lett. 35, 1578 (2010)
M. Stratmann, T. Pagel, F. Mitschke, Phys. Rev. Lett. 95, 143902 (2005)
G. Herink, F. Kurtz, B. Jalali et al., Science 356, 50 (2017)
F. Mitschke, A. Hause, C. Mahnke, Eur. Phys. J. 225, 245364 (2016)
O. Melchert, S. Willms, S. Bose, A. Yulin, B. Roth, F. Mitschke, A. Demircan, Phys. Rev. Lett. 123, 243905 (2019)
G. Xu, A. Gelash, A. Chabchoub et al., Phys. Rev. Lett. 122, 084101 (2019)
M. Kirane, S. Stalin, R. Arun, M. Lakshmanan, arXiv:2308.16535 [nlin.PS] (2023)
B. Wang, H. Han, L. Yu et al., Nanophotonics 11, 129 (2021)
S.Y. Lou, J. Phys. Commun. 4, 041002 (2020)
M. Jia, J. Lin, S.Y. Lou, Nonlinear Dyn. 100, 3745 (2020)
X. Yang, R. Fan, B. Li, Phys. Scr. 95, 045213 (2020)
S.N. Lin, Y. Chen, J. Comput. Phys. 457, 111053 (2022)
L.H. Wang, K. Porsezian, J.S. He, Phys. Rev. E 87, 053202 (2013)
M.R. Ali, M.A. Khattab, S.M. Mabrouk, Optik 272, 170256 (2023)
B.Q. Li, Y.L. Ma, Nonlinear Dyn. 111, 6689–6699 (2023)
R. Hirota, Phys. Rev. Lett. 27, 1192 (1971)
W.T. Li, Z. Zhang, X.Y. Yang, B. Li, Int. J. Mod. Phys. B 33, 1950255 (2019)
B. Ren, J. Lin, Wave Motion 117, 103110 (2022)
B. Ren, J. Lin, Eur. Phys. J. Plus 136, 123 (2021)
Z. Yan, S.Y. Lou, Appl. Math. Lett. 104, 106271 (2020)
Z. Zhang, Q. Guo, B. Li, J. Chen, Commun. Nonlinear Sci. Numer. Simul. 101, 105866 (2021)
Y. Li, R. Yao, Y. Xia, S. Lou, Commun. Nonlinear Sci. Numer. Simul. 100, 105843 (2021)
P. Rohrmann, A. Hause, F. Mitschke, Phys. Rev. A 87, 043834 (2013)
A. Chowdury, D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Phys. Rev. E 91, 032928 (2015)
C. Mahnke, F. Mitschke, Phys. Rev. A 85, 033808 (2012)
P.F. Wei, C.X. Long, C. Zhu et al., Chaos Soliton Fract. 158, 112062 (2022)
C.Q. Dai, Y.Y. Wang, J.F. Zhang, Nonlinear Dyn. 102, 379 (2020)
X.W. Jin, J. Lin, J. Magn. Magn. Mater. 502, 166590 (2020)
Acknowledgements
This work is supported by the National Natural Science Foundation of China Grant Nos. 12375006, 11975156, 12105243 and 11775146. The authors are in debt to thank Dr. Xin-Wei Jin for his help to offer the numerical simulation on the stability of the soliton molecule.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Xiong, Z., Ren, B. & Wang, W. Soliton molecules, multi-breathers and dynamical behaviors of the Lakshmanan–Porsezian–Daniel equation. Eur. Phys. J. Plus 138, 1051 (2023). https://doi.org/10.1140/epjp/s13360-023-04682-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-023-04682-y