Skip to main content
SpringerLink
Log in

Modulation instability in nonlinear acoustic metamaterials with coupling coefficients

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

In this paper, the modulation instability is analyzed in a novel model of coupled pairs of chains with coupling coefficient parameters. We derive the coupled system of the nonlinear equation by using the Lindstedt–Poincaré perturbation and the multi-scale method. From the linear stability analysis, an expression for the modulation instability growth rate is established, and we assessed the effects of the coupling strength on the modulation instability spectrum in acoustic and optical modes. In acoustic metamaterials, it appears that modulation instability develops and results in the formation of wave trains. We also demonstrated that the modulation instability bandwidth can be reduced or expanded depending on the effectiveness of the coupling coefficient as well as the angular frequency. Through the numerical experiments, we showed the propagation of the modulated wave patterns, manifesting the existence of the modulation instability growth rate in the structure. It results from this investigation that conventional materials where microstructural configurations are designed to exhibit unusual properties are attractive to modulated wave structures. In future works, soliton dynamics, rogue waves, and localized modes will be investigated to give rise to several applications.

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
Fig. 9

Similar content being viewed by others

Availability of data and materials

Not applicable.

References

  1. G.L. Huang, W.J. Zhou, X.P. Li, B.Y.S. Wang, C.W.Q. Chen, Spectro-spatial analysis of wave packet propagation in nonlinear acoustic metamaterials. J. Sound Vib. 413, 25–69 (2018)

    Google Scholar 

  2. B. Deng, J.R. Raney, K. Bertoldi, V. Tournat, Nonlinear waves in flexible mechanical metamaterials. J. Appl. Phys. 130, 040901–8 (2021)

    Article  Google Scholar 

  3. Y. Ma, G. Dong, M. Perlin, X. Ma, G. Wang, Experimental investigation on the evolution of the modulation instability with dissipation. J. Fluid Mech. 711, 101–121 (2012)

    Article  ADS  MATH  Google Scholar 

  4. D. Solli, C. Ropers, P. Koonath, B. Jalali, Optical rogue waves. Nature 450, 1054–1057 (2007)

    Article  ADS  Google Scholar 

  5. S.A. Cummer, J. Christensen, A. Alù, Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016)

    Article  ADS  Google Scholar 

  6. J.M. Dudley, G. Genty, F. Dias, B. Kibler, N. Akhmediev, Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation. Opt. Exp. 17, 21497–21598 (2009)

    Article  Google Scholar 

  7. F. Copie, S. Randoux, P. Sure, The Physics of the one-dimensional nonlinear Schrödinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena. Rev. Phys. 5, 100037 (2020)

    Article  Google Scholar 

  8. K. Manktelow, M.J. Leamy, M. Ruzzene, Multiple scales analysis of wave- wave interactions in a cubically nonlinear monoatomic chain. Nonlinear Dyn. 63, 193–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Akinyemi, M. Mirzazadeh, K. Hosseini, Solitons and other solutions of perturbed nonlinear Biswas-Milovic equation with Kudryashov’s law of refractive index. Nonlinear Anal.: Modell. Control 27(3), 479–495 (2022)

    MathSciNet  MATH  Google Scholar 

  10. J. Mora, J. Mibaile, D. Vroumsia, A.S. Sylvere, B. Gambo, Modulational instability and rogue waves in one-dimensional nonlinear acoustic metamaterials: case of diatomic model. Phys. Scr. 96, 125274 (2021)

    Article  Google Scholar 

  11. A. Houwe, A. Souleymanou, Y. Saliou, L. Akinyemi, S.Y. Doka, Modulation instability gain and wave patterns in birefringent fibers induced by coupled nonlinear Schrödinger equation. Wave Motion 118, 103111 (2023)

    Article  MATH  Google Scholar 

  12. J. Xie, Z. Deng, X. Chang, B. Tang, Discrete modulational instability and bright localized spin wave modes in easy-axis weak ferromagnetic spin chains involving the next-nearest-neighbor coupling. Chin. Phys. B 28, 077501 (2019)

    Article  ADS  Google Scholar 

  13. G. Huang, Soliton excitations in one-dimensional diatomic lattices. Phys. Rev. B 51, 12347–60 (1995)

    Article  ADS  Google Scholar 

  14. A. Houwe, A. Souleymanou, L. Akinyemi, M. Inc, S.Y. Doka, Wave propagation in discrete cold bosonic atoms zig-zag optical lattice. Eur. Phys. J. Plus 137, 1029 (2022)

    Article  Google Scholar 

  15. F.K. Abdullaev, A. Bouketir, A. Messikh, B.A. Umarov, Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrödinger equation. Phys. D 232, 54 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. J.E. Rothenberg, Modulational instability for normal dispersion. Phys. Rev. A 42, 682 (1990)

    Article  ADS  Google Scholar 

  17. A. Souleymanou, A. Houwe, L. Akinyemi, Y. Saliou, T.B. Bouetou, Modulation instability gain and discrete soliton interaction in gyrotropic molecular chain. Chaos, Solitons Fractals 160, 112255 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  18. N. Akhmediev, V. Korneev, Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 69, 1089–1093 (1986)

    Article  MATH  Google Scholar 

  19. G. Van Simaeys, P. Emplit, M. Haelterman, Experimental study of the reversible behavior of modulational instability in optical fibers. J. Opt. Soc. Am. B 19, 477–486 (2002)

    Article  ADS  Google Scholar 

  20. F. Bonnefoy, A. Tikan, F. Copie, P. Suret, G. Ducrozet, G. Prabhudesai, G. Michel, A. Cazaubiel, E. Falcon, G. El, S. Randoux, From modulational instability to focusing dam breaks in water waves. Phys. Rev. F 5, 034802 (2020)

    Google Scholar 

  21. K. Tai, A. Hasegawa, A. Tomita, Observation of modnlational instability in optical fibers. Phy. Rev. Lett. 56, 135–138 (1986)

    Article  ADS  Google Scholar 

  22. J. Xie, Z. Deng, X. Chang, B. Tang, Discrete modulational instability and bright localized spin wave modes in easy-axis weak ferromagnetic spin chains involving the next-nearest-neighbor coupling. Chin. Phys. B 28, 077501 (2019)

    Article  ADS  Google Scholar 

  23. J.P. Nguenang, M. Peyrard, A.J. Kenfack, T.C. Kofane, On modulational instability of nonlinear waves in 1D ferromagnetic spin chains. J. Phys.: Condens. Matter 17, 3083 (2005)

    ADS  Google Scholar 

  24. L. Kavitha, E. Parasuraman, D. Gopi, A. Prabhu, R.A. Vicencio, Nonlinear nano-scale localized breather modes in a discrete weak ferromagnetic spin lattice. J. Magn. Magn. Mater. 401, 394 (2016)

    Article  ADS  Google Scholar 

  25. S.G. Murdoch, R. Leonhardt, J.D. Harvey, Polarization modulation instability in weakly birefringent fibers. Opt. Lett. 20, 866 (1995)

    Article  ADS  Google Scholar 

  26. T. Tanemura, Y. Ozeki, K. Kikuchi, Modulational instability and parametric amplification induced by loss dispersion in optical fibers. Phys. Rev. Lett. 93, 163902 (2004)

    Article  ADS  Google Scholar 

  27. T.B. Benjamin, J.E. Feir, The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430 (1967)

    Article  ADS  MATH  Google Scholar 

  28. A. Chabchoub, N.P. Hoffmann, N. Akhmediev, Rogue wave observation in a water wave tank. PRL. 106, 204502 (2011)

    Article  ADS  Google Scholar 

  29. B. Félicien, T. Alexey, C. François, S. Pierre, D. Guillaume, P. Gaurav, M. Guillaume, C. Annette, F. Eric, E. Gennady, R. Stéphane, From modulational instability to focusing dam breaks in water waves. Phys. Rev. Fluids 5, 034802 (2020)

    Article  Google Scholar 

  30. C. Mo, J. Singh, J.R. Raney, P.K. Purohit, Cnoidal wave propagation in an elastic metamaterial. Phys. Rev. E 100, 013001 (2019)

    Article  ADS  Google Scholar 

  31. B. Deng, J.R. Raney, V. Tournat, K. Bertoldi, Elastic vector solitons in soft architected materials. Phys. Rev. Lett. 118, 204102 (2017)

    Article  ADS  Google Scholar 

  32. A. Demiquel, V. Achilleos, G. Theocharis, V. Tournat, Modulation instability in nonlinear flexible mechanical metamaterials. Phys. Rev. E 107, 054212 (2023)

    Article  ADS  Google Scholar 

Download references

Funding

No funding available for this project.

Author information

Authors and Affiliations

  1. Department of Marine Engineering, Limbé Nautical Arts and Fisheries Institute, P.O Box 485, Limbé, Cameroon

    Alphonse Houwe

  2. Department of Basic Science, National Advanced School of Mines and Petroleum Industries, The University of Maroua, P.O Box 08, kaélé, Cameroon

    Souleymanou Abbagari & Serge Yamigno Doka

  3. Department of Mathematics, Lafayette College, Easton, Pennsylvania, USA

    Lanre Akinyemi

  4. Department of Mathematics, Faculty of Science, Firat University, 23119, Elazig, Turkey

    Mustafa Inc

  5. Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

    Mustafa Inc

  6. Department of Physics, Faculty of Science, The University of Ngaoundèré, P.O Box 454, Ngaoundèré, Cameroon

    Serge Yamigno Doka

Authors
  1. Alphonse Houwe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Souleymanou Abbagari
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lanre Akinyemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mustafa Inc
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Serge Yamigno Doka
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Alphonse Houwe, Souleymanou Abbagari or Lanre Akinyemi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Ethics approval

Not applicable

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Houwe, A., Abbagari, S., Akinyemi, L. et al. Modulation instability in nonlinear acoustic metamaterials with coupling coefficients. Eur. Phys. J. Plus 138, 560 (2023). https://doi.org/10.1140/epjp/s13360-023-04195-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04195-8

Search

Navigation