Skip to main content
SpringerLink
Log in

Conservation laws, periodic and rational solutions for an extended modified Korteweg–de Vries equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We study an integrable extended modified Korteweg–de Vries equation, which contains the fifth-order dispersion and relevant higher-order nonlinear terms. The infinitely many conservation laws are constructed based on the Lax pair. A general Nth-order periodic solution is obtained by means of the N-fold Darboux transformation (DT), and a simple representation of the Nth-order rational solution is derived from the generalized DT by using the limit approach. As an application, the explicit periodic and rational solutions from first to second order are given, and some typical nonlinear wave patterns such as the doubly periodic lattice-like and doubly localized high-peak waves are shown. It is interestingly found that, the doubly localized high-peak wave can be converted into a W-shaped soliton in the second-order rational solution due to the existence of higher-order terms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Wadati, M.: The modified Korteweg–de Vries equation. J. Phys. Soc. Jpn. 34, 1289 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Leblond, H., Grelu, Ph, Mihalache, D.: Models for supercontinuum generation beyond the slowly-varying-envelope approximation. Phys. Rev. A 90, 053816 (2014)

    Article  Google Scholar 

  3. Ono, H.: Soliton fission in anharmonic lattices with reflectionless inhomogeneity. J. Phys. Soc. Jpn. 61, 4336 (1992)

    Article  Google Scholar 

  4. Khater, A.H., El-Kalaawy, O.H., Callebaut, D.K.: Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron–positron plasma. Phys. Scr. 58, 545 (1998)

    Article  Google Scholar 

  5. Lonngren, K.E.: Ion acoustic soliton experiments in a plasma. Opt. Quantum Electron. 30, 615 (1998)

    Article  Google Scholar 

  6. Helal, M.A.: Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos Soliton Fractals 13, 1917 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hirota, R.: Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gu, C.H., Zhou, Z.X.: On the Darboux matrices of Bäcklund transformations for AKNS systems. Lett. Math. Phys. 13, 179 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675 (2009)

    Article  MATH  Google Scholar 

  10. Garrett, C., Gemmrich, J.: Rogue waves. Phys. Today 62, 62 (2009)

    Article  Google Scholar 

  11. Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054 (2007)

    Article  Google Scholar 

  12. He, J.S., Charalampidis, E.G., Kevrekidis, P.G., Frantzeskakis, D.J.: Rogue waves in nonlinear Schrödinger models with variable coefficients: application to Bose–Einstein condensates. Phys. Lett. A 378, 577 (2014)

    Article  MATH  Google Scholar 

  13. Moslem, W.M., Sabry, R., El-Labany, S.K., Shukla, P.K.: Dust-acoustic rogue waves in a nonextensive plasma. Phys. Rev. E 84, 066402 (2011)

    Article  Google Scholar 

  14. Yan, Z.Y.: Vector financial rogue waves. Phys. Lett. A 375, 4274 (2011)

    Article  MATH  Google Scholar 

  15. Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. B Appl. Math. 25, 16 (1983)

    Article  MATH  Google Scholar 

  16. Akhmediev, N., Korneev, V.I.: Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69, 1089 (1986)

    Article  MATH  Google Scholar 

  17. Kuznetsov, E.A.: Solitons in a parametrically unstable plasma. Sov. Phys. Dokl. 22, 507 (1977)

    Google Scholar 

  18. Ma, Y.C.: The Perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math. 60, 43 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhang, D.J., Zhao, S.L., Sun, Y.Y., Zhou, J.: Solutions to the modified Korteweg–de Vries equation. Rev. Math. Phys. 26, 1430006 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chowdury, A., Ankiewicz, A., Akhmediev, N.: Periodic and rational solutions of modified Korteweg–de Vries equation. Eur. Phys. J. D 70, 104 (2016)

    Article  Google Scholar 

  21. Slunyaev, A.V., Pelinovsky, E.N.: Role of multiple soliton interactions in the generation of rogue waves: the modified Korteweg–de Vries framework. Phys. Rev. Lett. 117, 214501 (2016)

    Article  Google Scholar 

  22. Xing, Q.X., Wang, L.H., Mihalache, D., Porsezian, K., He, J.S.: Construction of rational solutions of the real modified Korteweg–de Vries equation from its periodic solutions. Chaos 27, 053102 (2017)

    Article  MathSciNet  Google Scholar 

  23. Marchant, T.R.: Asymptotic solitons for a higher-order modified Korteweg–de Vries equation. Phys. Rev. E 66, 046623 (2002)

    Article  MathSciNet  Google Scholar 

  24. Tao, Y.S., He, J.S.: Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation. Phys. Rev. E 85, 026601 (2012)

    Article  Google Scholar 

  25. Wang, X., Li, Y.Q., Chen, Y.: Generalized Darboux transformation and localized waves in coupled Hirota equations. Wave Motion 51, 1149 (2014)

    Article  MathSciNet  Google Scholar 

  26. Wang, X., Liu, C., Wang, L.: Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations. J. Math. Anal. Appl. 449, 1534 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chen, S.H.: Twisted rogue-wave pairs in the Sasa–Satsuma equation. Phys. Rev. E 88, 023202 (2013)

    Article  Google Scholar 

  28. Chowdury, A., Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy. Phys. Rev. E 91, 032928 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Liu, C., Yang, Z.Y., Zhao, L.C., Yang, W.L.: State transition induced by higher-order effects and background frequency. Phys. Rev. E 91, 022904 (2015)

    Article  Google Scholar 

  30. Wang, L., Zhang, J.H., Liu, C., Li, M., Qi, F.H.: Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Phys. Rev. E 93, 062217 (2016)

    Article  MathSciNet  Google Scholar 

  31. Wang, L., Zhang, J.H., Liu, C., Li, M., Qi, F.H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016)

    Article  MathSciNet  Google Scholar 

  32. Zhang, J.H., Wang, L., Liu, C.: Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects. Proc. R. Soc. A 473, 20160681 (2017)

    Article  MathSciNet  Google Scholar 

  33. Wang, X., Liu, C., Wang, L.: Rogue waves and W-shaped solitons in the multiple self-induced transparency system. Chaos 27, 093106 (2017)

    Article  MathSciNet  Google Scholar 

  34. Wazwaz, A.M., Xu, G.Q.: An extended modified KdV equation and its Painlevé integrability. Nonlinear Dyn. 86, 1455 (2016)

    Article  MATH  Google Scholar 

  35. Wei, J., Geng, X.G.: A vector generalization of Volterra type differential–difference equations. Appl. Math. Lett. 55, 36 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wei, J., Geng, X.G.: A hierarchy of new nonlinear evolution equations and generalized bi-Hamiltonian structures. Appl. Math. Comput. 268, 664 (2015)

    MathSciNet  Google Scholar 

  37. Wei, J., Geng, X.G., Zeng, X.: Quasi-periodic solutions to the hierarchy of four-component Toda lattices. J. Geom. Phys. 106, 26 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  Google Scholar 

  39. Baronio, F., Degasperis, A., Conforti, M., Wabnitz, S.: Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109, 044102 (2012)

    Article  Google Scholar 

  40. Xu, S.W., He, J.S.: The Darboux transformation of the derivative nonlinear Schrödinger equation. J. Phys. A 44, 305203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, X., Li, Y.Q., Huang, F., Chen, Y.: Rogue wave solutions of AB system. Commun. Nonlinear Sci. Numer. Simul. 20, 434 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, L., Wang, Z.Q., Sun, W.R., Shi, Y.Y., Li, M., Xu, M.: Dynamics of Peregrine combs and Peregrine walls in an inhomogeneous Hirota and Maxwell–Bloch system. Commun. Nonlinear Sci. Numer. Simul. 47, 190 (2017)

    Article  MathSciNet  Google Scholar 

  43. Wang, L., Jiang, D.Y., Qi, F.H., Shi, Y.Y., Zhao, Y.C.: Dynamics of the higher-order rogue waves for a generalized mixed nonlinear Schrödinger model. Commun. Nonlinear Sci. Numer. Simul. 42, 502 (2017)

    Article  MathSciNet  Google Scholar 

  44. Zhang, X.E., Chen, Y.: Rogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo–Miwa equation. Commun. Nonlinear Sci. Numer. Simul. 52, 24 (2017)

    Article  MathSciNet  Google Scholar 

  45. Xu, T., Chen, Y.: Darboux transformation of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization. Commun. Nonlinear Sci. Numer. Simul. 57, 276 (2018)

    Article  MathSciNet  Google Scholar 

  46. Wei, J., Wang, X., Geng, X.G.: Periodic and rational solutions of the reduced Maxwell–Bloch equations. Commun. Nonlinear Sci. Numer. Simul. 59, 1 (2018)

    Article  MathSciNet  Google Scholar 

  47. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tiofack, C.G.L., Coulibaly, S., Taki, M., De Bievre, S., Dujardin, G.: Comb generation using multiple compression points of Peregrine rogue waves in periodically modulated nonlinear Schrödinger equations. Phys. Rev. A 92, 043837 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 11705290, 11671075, 11305060), China Postdoctoral Science Foundation funded sixtieth batches (Grant No. 2016M602252) and Key Research Projects of Henan Higher Education Institutions (Grant No. 18A110038).

Author information

Authors and Affiliations

  1. College of Science, Zhongyuan University of Technology, Zhengzhou, 450007, Henan, China

    Xin Wang & Jianlin Zhang

  2. School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, 450001, Henan, China

    Xin Wang

  3. School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

    Lei Wang

Authors
  1. Xin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianlin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xin Wang.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Zhang, J. & Wang, L. Conservation laws, periodic and rational solutions for an extended modified Korteweg–de Vries equation. Nonlinear Dyn 92, 1507–1516 (2018). https://doi.org/10.1007/s11071-018-4143-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4143-z

Keywords

Search

Navigation