Skip to main content
SpringerLink
Log in

Rogue waves for a long wave–short wave resonance model with multiple short waves

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

Abstract

A resonance between long and short waves will occur if the phase velocity of the long wave matches the group velocity of the short wave. In this paper, a system with two distinct packets of short waves in resonance with a common long wave is studied. Breather solutions are calculated by the Hirota bilinear method, and rogue wave modes (unexpectedly large displacements from an otherwise calm background state) are obtained from the breathers through a long wave limit. The location and magnitude of the maximum displacement are determined quantitatively. Remarkably this coupling enables a rogue wave to attain a larger magnitude than that in a configuration with just one single short wave component. Furthermore, as the wavenumber varies, a transition from an elevation rogue wave to a depression rogue wave is possible. This transformation of the wave profile is elucidated in terms of the properties of the carrier envelope. The connection with the modulation instability of the background plane wave is investigated. Some numerical simulations are performed to demonstrate both the robust nature and unstable behavior for these rogue waves, depending on the parameters of the system. Dynamics and properties of rogue waves with three or more short wave components are also considered.

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
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Craik, A.D.D.: Wave Interactions and Fluid Flows. Cambridge University Press, New York (1985)

    MATH  Google Scholar 

  2. Kivshar, Y.S., Agrawal, G.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, San Diego (2003)

    Google Scholar 

  3. Benney, D.J.: Significant interactions between small and large scale surface waves. Stud. Appl. Math. 55, 93 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ma, Y.C.: The complete solution of the long-wave-short-wave resonance equations. Stud. Appl. Math. 59, 201 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ma, Y.C., Redekopp, L.G.: Some solutions pertaining to the resonant interaction of long and short waves. Phys. Fluids 22, 1872 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47 (2013)

    Article  MathSciNet  Google Scholar 

  7. Ivancevic, V.G., Reid, D.J.: Turbulence and shock waves in crowd dynamics. Nonlinear Dyn. 68, 285 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhu, H.P.: Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides. Nonlinear Dyn. 72, 873 (2013)

    Article  MathSciNet  Google Scholar 

  9. Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, 026601 (2009)

    Article  MATH  Google Scholar 

  10. Bludov, Yu.V., Konotop, V.V., Akhmediev, N.: Vector rogue waves in binary mixtures of Bose–Einstein condensates. Eur. Phys. J. Spec. Top. 185, 169 (2010)

  11. Vinayagam, P.S., Radha, R., Porsezian, K.: Taming rogue waves in vector Bose–Einstein condensates. Phys. Rev. E 88, 042906 (2013)

    Article  Google Scholar 

  12. Zhao, L.C., Liu, J.: Localized nonlinear waves in a two-mode nonlinear fiber. J. Opt. Soc. Am. B 29, 3119 (2012)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Zhai, B.G., Zhang, W.G., Wang, X.L., Zhang, H.Q.: Multi-rogue waves and rational solutions of the coupled nonlinear Schrödinger equations. Nonlinear Anal. Real World Appl. 14, 14 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chow, K.W., Chan, H.N., Kedziora, D.J., Grimshaw, R.H.J.: Rogue wave modes for the long wave-short wave resonance model. J. Phys. Soc. Jpn. 82, 074001 (2013)

    Article  Google Scholar 

  16. Matsuno, Y.: Bilinear Transformation Method. Academic Press, Orlando (1984)

    MATH  Google Scholar 

  17. Chen, S., Grelu, P., Soto-Crespo, J.M.: Dark- and bright-rogue-wave solutions for media with long-wave-short-wave resonance. Phys. Rev. E 89, 011201(R) (2014)

    Article  Google Scholar 

  18. Dudley, J.M., Genty, G., Dias, F., Kibler, B., Akhmediev, N.: Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation. Opt. Expr. 17, 21497 (2009)

    Article  Google Scholar 

  19. Ma, Y.C.: The resonant interaction among long and short waves. Wave Motion 3, 257 (1981)

    Article  MATH  Google Scholar 

  20. Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Classifying the hierarchy of nonlinear-Schrödinger-equation rogue wave solutions. Phys. Rev. E 88, 013207 (2013)

    Article  Google Scholar 

  21. Shrira, V.I., Geogjaev, V.V.: What makes the Peregrine soliton so special as a prototype of freak waves? J. Eng. Math. 67, 11 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sakkaravarthi, K., Kanna, T., Vijayajayanthi, M., Lakshmanan, M.: Multicomponent long-wave-short-wave resonance interaction system: bright solitons, energy-sharing collisions, and resonant solitons. Phys. Rev. E 90, 052912 (2014)

    Article  MATH  Google Scholar 

  23. Kanna, T., Vijayajayanthi, M., Sakkaravarthi, K., Lakshmanan, M.: Higher dimensional bright solitons and their collisions in a multicomponent long wave-short wave system. J. Phys. A Math. Theor. 42, 115103 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen, J., Chen, Y., Feng, B.-F., Maruno, K.: General mixed multi-soliton solutions to one-dimensional multicomponent Yajima–Oikawa system. J. Phys. Soc. Jpn. 84, 074001 (2015)

    Article  Google Scholar 

  25. Chen, J., Chen, Y., Feng, B.-F., Maruno, K.: Rational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systems. Phys. Lett. A 379, 1510 (2015)

    Article  MathSciNet  Google Scholar 

  26. Baronio, F., Conforti, M., Degasperis, A., Lombardo, S., Onorato, M., Wabnitz, S.: Vector rogue waves and baseband modulation instability in the defocusing regime. Phys. Rev. Lett. 113, 034101 (2014)

    Article  Google Scholar 

  27. Chen, S., Soto-Crespo, J.M., Grelu, P.: Coexisting rogue waves within the (2+1)-component long-wave-short-wave resonance. Phys. Rev. E 90, 033203 (2014)

    Article  Google Scholar 

  28. Li, J.H., Chan, H.N., Chiang, K.S., Chow, K.W.: Breathers and ‘black’ rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity of opposite signs. Commun. Nonlinear Sci. Numer. Simulat. 28, 28 (2015)

    Article  MathSciNet  Google Scholar 

  29. Chan, H.N., Malomed, B.A., Chow, K.W., Ding, E.: Rogue waves for a system of coupled derivative nonlinear Schrödinger equations. Phys. Rev. E 93, 012217 (2016)

    Article  Google Scholar 

  30. Kanna, T., Vijayajayanthi, M., Lakshmanan, M.: Mixed solitons in a (2+1)-dimensional multicomponent long-wave-short-wave system. Phys. Rev. E 90, 042901 (2014)

    Article  Google Scholar 

  31. 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 

Download references

Acknowledgments

D.J.K. acknowledges the support from the Australian Research Council (Discovery Project No. DP110102068). Partial financial support for the present research team has been provided by the Research Grants Council General Research Fund contracts HKU711713E and HKU17200815.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong

    Hiu Ning Chan & Kwok Wing Chow

  2. Department of Mathematics and Physics, Azusa Pacific University, Azusa, CA, 91702, USA

    Edwin Ding

  3. Research School of Physics and Engineering, Australian National University, Canberra, ACT, 0200, Australia

    David Jacob Kedziora

  4. Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

    Roger Grimshaw

Authors
  1. Hiu Ning Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Edwin Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Jacob Kedziora
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Roger Grimshaw
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Kwok Wing Chow
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kwok Wing Chow.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chan, H.N., Ding, E., Kedziora, D.J. et al. Rogue waves for a long wave–short wave resonance model with multiple short waves. Nonlinear Dyn 85, 2827–2841 (2016). https://doi.org/10.1007/s11071-016-2865-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2865-3

Keywords

Search

Navigation