Home The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations
Article
Licensed
Unlicensed Requires Authentication

The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

  • Wei Tan EMAIL logo and Zhao-Yang Yin
Published/Copyright: March 18, 2021
Become an author with De Gruyter Brill
Author Information
International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 22 Issue 3-4

Abstract

The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

Keywords: Hirota’s bilinear method; homoclinic solution; parameter limit method; rogue wave solution
PACS: 02.30.Ik; 02.30.Jr; 02.70.Wz
Corresponding author: Wei Tan, Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China; and College of Mathematics and Statistics, Jishou University, Jishou 416000, China, E-mail:

Acknowledgements

This work was supported by the National Natural Science Foundation of P.R. China (11661037) and the Scientific Research Project of Hunan Education Department (17C1297).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] Y. Ohta and J. Yang, “General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation,” Proc. R. Soc. A, vol. 468, pp. 1716–1740, 2011. https://doi.org/10.1098/rspa.2011.0640.Search in Google Scholar

[2] L. Draper, “Freak wave,” Mar. Obs., vol. 35, no. 2, pp. 193–195, 1965.Search in Google Scholar

[3] D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature, vol. 450, no. 7172, pp. 1054–1057, 2007. https://doi.org/10.1038/nature06402.Search in Google Scholar

[4] D. R. Solli, C. Ropers, and B. Jalali, “Active control of rogue waves for stimulated supercontinuum generation,” Phys. Rev. Lett., vol. 101, p. 233902, 2008. https://doi.org/10.1103/physrevlett.101.233902.Search in Google Scholar

[5] Z. Y. Yan, “Vector financial rogue waves,” Phys. Lett. A, vol. 375, pp. 4274–4279, 2011. https://doi.org/10.1016/j.physleta.2011.09.026.Search in Google Scholar

[6] M. Onoratoa, S. Residori, and U. Bortolozzoc, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep., vol. 528, pp. 47–89, 2013. https://doi.org/10.1016/j.physrep.2013.03.001.Search in Google Scholar

[7] L. X. Li and Z. D. Dai, “New homoclinic rogue wave solution for the coupled nonlinear Schrödinger–Boussinesq equation,” J. Nonlinear Sci. Appl., vol. 10, pp. 2642–2648, 2017. https://doi.org/10.22436/jnsa.010.05.30.Search in Google Scholar

[8] G. Mu and Z. Qin, “Rogue waves for the coupled Schrödinger-Boussinesq equation and the coupled Higgs equation,” J. Phys. Soc. Jpn., vol. 81, p. 4001, 2012. https://doi.org/10.1143/jpsj.81.084001.Search in Google Scholar

[9] Y. V. Bludov, V. V. Konotop, and N. Akhmediev, “Matter rogue waves,” Phys. Rev. A, vol. 80, pp. 2962–2964, 2009. https://doi.org/10.1103/physreva.80.033610.Search in Google Scholar

[10] D. H. Peregrine, “Water wave, nonlinear Schrödinger equation and their solution,” J. Austral. Math. Soc. Ser. B, vol. 25, pp. 16–43, 1983 https://doi.org/10.1017/s0334270000003891 .Search in Google Scholar

[11] L. X. Li, J. Liu, Z. D. Dai, and R. L. Liu, “New rational homoclinic and rogue wave solution for the coupled Schrödinger equation,” Z. Naturforsch. A, vol. 69, pp. 441–448, 2014. https://doi.org/10.5560/zna.2014-0039.Search in Google Scholar

[12] W. X. Ma, “Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations,” Appl. Math. Lett., vol. 102, p. 106161, 2020. https://doi.org/10.1016/j.aml.2019.106161.Search in Google Scholar

[13] M. S. Osman, “Nonlinear interaction of solitary waves described by multi-rational wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili equation with variable coefficients,” Nonlinear Dynam., vol. 87, pp. 1209–1216, 2017. https://doi.org/10.1007/s11071-016-3110-9.Search in Google Scholar

[14] W. Tan, H. P. Dai, Z. D. Dai, and W. Y. Zhong, “Emergence and space-time structure of lump solution to the (2 + 1)-dimensional generalized KP equation,” Pramana - J. Phys., vol. 89, pp. 77–84, 2017. https://doi.org/10.1007/s12043-017-1474-0.Search in Google Scholar

[15] W. X. Ma, “Lump solutions to the Kadomtsev–Petviashvili equation,” Phys. Lett. E, vol. 379, pp. 1975–1978, 2015. https://doi.org/10.1016/j.physleta.2015.06.061.Search in Google Scholar

[16] Y. Ohta and J. Yang, “Rogue waves in the Davey–Stewartson equation,” Phys. Rev. E, vol. 86, pp. 2386–2398, 2012. https://doi.org/10.1103/physreve.86.036604.Search in Google Scholar

[17] M. J. Ablowitz and J. Satsuma, “Solitons and rational solutions of nonlinear evolution equations,” J. Math. Phys., vol. 19, pp. 2180–2186, 1978. https://doi.org/10.1063/1.523550.Search in Google Scholar

[18] J. Satsuma, “Two-dimensional lumps in nonlinear dispersive systems,” J. Math. Phys., vol. 20, pp. 1496–1503, 1979. https://doi.org/10.1063/1.524208.Search in Google Scholar

[19] 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

[20] W. X. Ma and Y. Zhou, “Lump solutions to nonlinear partial differential equations via Hirota bilinear forms,” J. Differ. Equ., vol. 264, pp. 2633–2659, 2018. https://doi.org/10.1016/j.jde.2017.10.033.Search in Google Scholar

[21] W. X. Ma, C. X. Li, and J. S. He, “A second Wronskian formulation of the Boussinesq equation,” Nonlinear Anal. Theor. Methods Appl., vol. 70, no. 12, pp. 4245–4258, 2009. https://doi.org/10.1016/j.na.2008.09.010.Search in Google Scholar

[22] W. Tan, Z. D. Dai, and Z. Y. Yin, “Dynamics of multi-breathers, N-solitons and M-lump solutions in the (2 + 1)-dimensional KdV equation,” Nonlinear Dynam., vol. 96, pp. 1605–1614, 2019. https://doi.org/10.1007/s11071-019-04873-2.Search in Google Scholar

[23] C. J. Wang, “Lump solution and integrability for the associated Hirota bilinear equation,” Nonlinear Dynam., vol. 87, pp. 2635–2642, 2017. https://doi.org/10.1007/s11071-016-3216-0.Search in Google Scholar

[24] W. X. Ma, “Lump and interaction solutions to linear PDEs in 2 + 1 dimensions via symbolic computation,” Mod. Phys. Lett. B, vol. 33, p. 1950457, 2019. https://doi.org/10.1142/s0217984919504578.Search in Google Scholar

[25] A. J. Zhu and Z. D. Dai, “Homoclinic degeneracy for perturbed nonlinear Schrödinger equation,” Phys. Lett. A, vol. 363, pp. 102–107, 2007. https://doi.org/10.1016/j.physleta.2006.10.078.Search in Google Scholar

[26] W. X. Ma, “Direct search for exact solutions to the nonlinear Schrödinger equation,” Appl. Math. Comput., vol. 215, no. 8, pp. 2835–2842, 2009. https://doi.org/10.1016/j.amc.2009.09.024.Search in Google Scholar

[27] M. T. Darvishi, M. Najafi, and M. Najafi, “New exact solutions and Wronskian form of a (2 + 1)-dimensional potential Kadomtsev–Petviashvili equation,” Int. J. Nonlinear Sci., vol. 12, pp. 387–393, 2011.Search in Google Scholar

[28] X. P. Zeng, Z. D. Dai, and D. L. Li, “Some exact periodic soliton solutions and resonance for the potential Kadomtsev–Petviashvili equation,” J. Phys.: Conf. Ser., vol. 96, p. 012149, 2008. https://doi.org/10.1088/1742-6596/96/1/012149.Search in Google Scholar

[29] D. Kaya and S. M. El-Sayed, “Numerical soliton-like solutions of the potential Kadomtsev–Petviashvili equation by the decomposition method,” Phys. Lett. A, vol. 320, pp. 192–199, 2003. https://doi.org/10.1016/j.physleta.2003.11.021.Search in Google Scholar

[30] W. Tan and Z. D. Dai, “Dynamics of kinky wave for (3 + 1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation,” Nonlinear Dynam., vol. 85, pp. 817–823, 2016. https://doi.org/10.1007/s11071-016-2725-1.Search in Google Scholar

[31] C. J. Wang, Z. D. Dai, and C. F. Liu, “Interaction between kink solitary wave and rogue wave for (2 + 1)-dimensional Burgers equation,” Mediterr. J. Math., vol. 13, pp. 1087–1098, 2016. https://doi.org/10.1007/s00009-015-0528-0.Search in Google Scholar

[32] W. Tan, Z. D. Dai, J. L. Xie, and L. L. Hu, “Emergence and interaction of the lump-type solution with the (3 + 1)D Jimbo–Miwa equation,” Z. Naturforsch. A, vol. 73, pp. 43–50, 2018. https://doi.org/10.1515/zna-2017-0293.Search in Google Scholar

[33] W. X. Ma, Z. Qin, and L. Xing, “Lump solutions to dimensionally reduced p-gKP and p-BKP equations,” Nonlinear Dynam., vol. 84, pp. 923–931, 2016. https://doi.org/10.1007/s11071-015-2539-6.Search in Google Scholar

[34] W. X. Ma and L. Q. Zhang, “Lump solutions with higher-order rational dispersion relations,” Pramana - J. Phys., vol. 94, p. 43, 2020. https://doi.org/10.1007/s12043-020-1918-9.Search in Google Scholar

[35] X. G. Geng, “Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations,” J. Phys. A Gen. Phys., vol. 36, pp. 2289–2303, 2003. https://doi.org/10.1088/0305-4470/36/9/307.Search in Google Scholar

[36] Z. Q. Lao, “Rogue waves and rational solutions of a (3 + 1)-dimensional nonlinear evolution equation,” Phys. Lett. A, vol. 377, no. 42, pp. 3021–3026, 2013. https://doi.org/10.1016/j.physleta.2013.09.023.Search in Google Scholar

[37] Y. Zhang, Y. P. Liu, and X. Y. Tang, “M-lump solutions to a (3 + 1)-dimensional nonlinear evolution equation,” Comput. Math. Appl., vol. 76, pp. 592–601, 2018. https://doi.org/10.1016/j.camwa.2018.04.039.Search in Google Scholar

[38] G. Mu, Z. D. Dai, and Z. H. Zhao, “Localized structures for (2 + 1)-dimensional Boiti–Leon–Pempinelli equation,” Pramana - J. Phys., vol. 81, pp. 367–376, 2013. https://doi.org/10.1007/s12043-013-0577-5.Search in Google Scholar

[39] W. X. Ma and Y. J. Zhang, “Darboux transformations of integrable couplings and applications,” Rev. Math. Phys., vol. 30, p. 1850003, 2018. https://doi.org/10.1142/s0129055x18500034.Search in Google Scholar

[40] W. Tan, Z. D. Dai, J. L. Xie, and D. Q. Qiu, “Parameter limit method and its application in the (4 + 1)-dimensional Fokas equation,” Comput. Math. Appl., vol. 75, pp. 4214–4220, 2018. https://doi.org/10.1016/j.camwa.2018.03.023.Search in Google Scholar

[41] H. P. Dai and W. Tan, “Deformation characteristics of three-wave solutions and lump N-solitons to the (2 + 1)-dimensional generalized KdV equation,” Eur. Phys. J. Plus, vol. 135, no. 2, p. 239, 2020. https://doi.org/10.1140/epjp/s13360-020-00233-x.Search in Google Scholar

[42] W. Tan, W. Zhang, and J. Zhang, “Evolutionary behavior ofbreathers and interaction solutions with M-solitons for (2 + 1)-dimensional KdV system,” Appl. Math. Lett., vol. 101, p. 106063, 2020. https://doi.org/10.1016/j.aml.2019.106063.Search in Google Scholar

[43] S. C. Mancas, H. C. Rosu, and M. Pere-Maldonado, “Travelling-wave solutions for wave equations with two exponential nonlinearities,” Z. Naturforsch. A, vol. 73, pp. 883–892, 2018. https://doi.org/10.1515/zna-2018-0055.Search in Google Scholar

[44] J. Q. Lü, S. Bilige, and X. Q. Gao, “Abundant lump solution and interaction phenomenon of (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 20, no. 1, pp. 33–40, 2019. https://doi.org/10.1515/ijnsns-2018-0034.Search in Google Scholar

[45] Ö. Ünsal, W. X. Ma, and Y. J. Zhang, “Multiple-wave solutions to generalized bilinear equations in terms of hyperbolic and trigonometric solutions,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 18, pp. 395–401, 2017. https://doi.org/10.1515/ijnsns-2015-0127.Search in Google Scholar

[46] A. M. Wazwaz, “A new integrable equation that combines the KdV equation with the negative-order KdV equation,” Math. Methods Appl. Sci., vol. 41, pp. 80–87, 2018. https://doi.org/10.1002/mma.4595.Search in Google Scholar

[47] W. X. Ma, “Lump-type solutions to the (3 + 1)-dimensional Jimbo–Miwa equation,” Int. J. Nonlinear Sci. Numer. Simul., vol. 17, pp. 355–359, 2016. https://doi.org/10.1515/ijnsns-2015-0050.Search in Google Scholar

[48] W. X. Ma and E. G. Fan, “Linear superposition principle applying to Hirota bilinear equations,” Comput. Math. Appl., vol. 61, no. 4, pp. 950–959, 2011. https://doi.org/10.1016/j.camwa.2010.12.043.Search in Google Scholar

[49] W. X. Ma, “Interaction solutions to Hirota–Satsuma–Ito equation in (2 + 1)-dimensions,” Front. Math. China, vol. 14, pp. 619–629, 2019. https://doi.org/10.1007/s11464-019-0771-y.Search in Google Scholar

[50] S. J. Chen, Y. H. Yin, W. X. Ma, and X. Lü, “Abundant exact solutions and interaction phenomena of the (2 + 1)-dimensional YTSF equation,” Anal. Math. Phys., vol. 9, pp. 2329–2344, 2019. https://doi.org/10.1007/s13324-019-00338-2.Search in Google Scholar

[51] S. J. Chen, W. X. Ma, and X. Lü, “Böcklund transformation, exact solutions and interaction behaviour of the (3 + 1)-dimensional Hirota–Satsuma–Ito-like equation,” Commun. Nonlinear Sci. Numer. Simulat., vol. 83, p. 105135, 2020. https://doi.org/10.1016/j.cnsns.2019.105135.Search in Google Scholar

[52] J. Y. Yang and W. X. Ma, “Abundant interaction solutions of the KP equation,” Nonlinear Dynam., vol. 89, pp. 1539–1544, 2017. https://doi.org/10.1007/s11071-017-3533-y.Search in Google Scholar
Received: 2018-12-04
Revised: 2020-11-29
Accepted: 2021-02-22
Published Online: 2021-03-18
Published in Print: 2021-06-25

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Numerical analysis of geometrical nonlinear aeroelasticity with CFD/CSD method
  4. Application of moving least squares algorithm for solving systems of Volterra integral equations
  5. Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative
  6. Stochastic models with multiplicative noise for economic inequality and mobility
  7. Fast interaction of Cu2+ with S2O3 2− in aqueous solution
  8. Rotorcraft fuselage/main rotor coupling dynamics modelling and analysis
  9. Modeling and fault identification of the gear tooth surface wear failure system
  10. Wavelet-optimized compact finite difference method for convection–diffusion equations
  11. A methodology for dynamic behavior analysis of the slider-crank mechanism considering clearance joint
  12. Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps
  13. The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations
  14. Distributed consensus of multi-agent systems with increased convergence rate
  15. Studies on population balance equation involving aggregation and growth terms via symmetries
  16. Bifurcation, routes to chaos, and synchronized chaos of electromagnetic valve train in camless engines
  17. Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation
  18. Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator
https://doi.org/10.1515/ijnsns-2018-0365
Keywords for this article
Hirota’s bilinear method; homoclinic solution; parameter limit method; rogue wave solution

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Numerical analysis of geometrical nonlinear aeroelasticity with CFD/CSD method
  4. Application of moving least squares algorithm for solving systems of Volterra integral equations
  5. Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative
  6. Stochastic models with multiplicative noise for economic inequality and mobility
  7. Fast interaction of Cu2+ with S2O3 2− in aqueous solution
  8. Rotorcraft fuselage/main rotor coupling dynamics modelling and analysis
  9. Modeling and fault identification of the gear tooth surface wear failure system
  10. Wavelet-optimized compact finite difference method for convection–diffusion equations
  11. A methodology for dynamic behavior analysis of the slider-crank mechanism considering clearance joint
  12. Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps
  13. The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations
  14. Distributed consensus of multi-agent systems with increased convergence rate
  15. Studies on population balance equation involving aggregation and growth terms via symmetries
  16. Bifurcation, routes to chaos, and synchronized chaos of electromagnetic valve train in camless engines
  17. Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation
  18. Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 23.11.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2018-0365/html
Scroll to top button