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Modulational instability in transversely connected nonlinear pendulum pairs

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Abstract

In this work, we investigate the modulational instability (MI) phenomenon in a chain of coupled pendulum pairs, where each pendulum is connected to the nearest neighbors in the longitudinal and transverse directions. Based on the obtained equation describing the dynamics of the model, we derive the coupled discrete nonlinear Schrödinger equation using the multiple scale method. We use the obtained coupled discrete nonlinear Schrödinger equation to study the possibility of modulational instability. The linear stability analysis leads us to obtain the growth rate of the MI. It reveals that the instability growth rate and MI band are dramatically affected by the transverse coupling parameter. Finally, we use the MI analysis to study the dynamics of the generated unstable plane wave solutions numerically. This confirms that the existence of MI in the lattice leads to the breakup of wave into periodic localized pulses which have the shape of soliton-like objects.

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References

  1. T.B. Benjamin, J.E. Feir, The disintegration of wave trains on deep water. Part 1 Theory. J. Fluid Mech. 27, 417–430 (1967)

    Article  ADS  MATH  Google Scholar 

  2. N.F. Piliptetskii, A.R. Rustamov, Observation of self-focusing of light in liquids. JETP Lett. 2, 55–56 (1965)

    Google Scholar 

  3. V.I. Bespalov, V.I. Talanov, Filamentary structure of light beams in nonlinear liquids. ZhETF Pisma Redaktsiiu 2, 471–476 (1966)

    ADS  Google Scholar 

  4. P. Marquié, J.M. Bilbault, M. Remoissenet, Generation of envelope and hole solitons in an experimental transmission line. Phys. Rev. E 49, 828 (1994)

    Article  ADS  Google Scholar 

  5. P. Marquié, J.M. Bilbault, M. Remoissenet, Observation of nonlinear localized modes in an electrical lattice. Phys. Rev. E 51, 6127 (1995)

    Article  ADS  Google Scholar 

  6. M. Remoissenet, Waves Called Solitons Concepts and Experiments (Springer, Berlin, 1999)

    Book  MATH  Google Scholar 

  7. A. Kenfack-Jiotsa, E. Tala-Tebue, Effect of Second-Neighbor Inductive Coupling on the Modulational Instability in a Coupled Line of Transmission. J. Phys. Soc. Jpn. 80, 034003 (2011)

    Article  ADS  Google Scholar 

  8. F. I.I. Ndzana, A. Mohamadou, T.C. Kofane, Modulated waves and chaotic-like behaviours in the discrete electrical transmission line. J. Phys. D: Apply. Phys. 40, 3254 (2007)

    Article  ADS  Google Scholar 

  9. R. Stearrett, L.Q. English, Experimental generation of intrinsic localized modes in a discrete electrical transmission line. J. Phys. D: Apply. Phys 40, 5394 (2007)

    Article  ADS  Google Scholar 

  10. Y. Doi, A. Nakatani, K. Yoshimura, Modulational instability of zone boundary mode and band edge modes in nonlinear diatomic lattices. Phys. Rev. E 79, 026603 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  11. B. Sadjo, C.B. Tabi, H. Edongue, A. Mohamadou, Coupled energy patterns in zigzag molecular chains. Wave Motion 17, 30052–5 (2017)

    MathSciNet  MATH  Google Scholar 

  12. S. Abbagari, A. Houwe, L. Akinyemi, Y. Saliou, T.B. Bouetou, Modulation instability gain and discrete soliton interaction in gyrotropic molecular chain. Chaos Solit. Fract. 160, 112255 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Abouem, A. Ribama, Z.I. Djoufack, J.P. Nguenang, Breather-impurity interactions and modulational instability in a quantum 2D Klein-Gordon chain. Eur. Phys. J. B 95, 86 (2022)

    Article  ADS  Google Scholar 

  14. E. Tala-Tebue, G. Roger Deffo, S.B. Yamgoue, A. Kenfack-Jiotsa, F.B. Pelap, Monoatomic chain: modulational instability and exact traveling wave solutions. Eur. Phys. J. Plus 135, 715 (2020)

    Article  Google Scholar 

  15. F. Gounoko Mounouna, E. Wamba, A.S. Tchakoutio Nguetcho, I.A. Bhat, J.M. Bilbault, Modulational stability brought by cubic-quartic interactions of the nearest-neighbor in FK model subjected in a parametrized on-site potential. Commun. Nonlinear Sci. Numer. Simulat. 105, 106088 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Ourabah, T. Yamano, Nonlinear Schrödinger equations involved in dark matter halos: modulational instability. Eur. Phys. J. Plus 135, 634 (2020)

    Article  Google Scholar 

  17. G. Fongang Achu, F.M. Moukam Kakmeni, A.M. Dikande, Breathing pulses in the damped-soliton model for nerves,. Phys. Rev. E 97, 012211 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  18. A.S. Foualeng Kamga, G. Fongang Achu, F.M. Moukam Kakmeni, P. Guemkam Ghomsi, F.T. Ndjomatchoua, C. Tchawoua, Continuous signalling pathways instability in an electromechanical coupled model for biomembranes and nerves. Eur. Phys. J. B 95, 12 (2022)

    Article  ADS  Google Scholar 

  19. M. Khalid, F. Hadi, A. ur Rahma, Modulation of multi-dimensional waves in anisotropic magnetized plasma. Eur. Phys. J. Plus 136, 1061 (2021)

    Article  Google Scholar 

  20. V.K. Sharma, Chirped soliton-like solutions of generalized nonlinear Schrödinger equation for pulse propagation in negative index material embedded into a Kerr medium. Indian J. Phys. 90, 1271 (2016)

    Article  ADS  Google Scholar 

  21. C.D. Bansi Kamdem, C.B. Tabi, A. Mohamadou, Dissipative Mayer’s waves in fluid-filled viscoelastic tubes. Chaos Solit. Fract. 109, 170 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463 (1999)

    Article  ADS  Google Scholar 

  23. F.S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni, M. Inguscio, Superfluid current disruption in a chain of weakly coupled Bose-Einstein condensates. New. J. Phys. 5, 71 (2003)

    Article  ADS  Google Scholar 

  24. B. Wu, Q. Nu, Landau and dynamical instabilities of the superflow of Bose-Einstein condensates in optical lattices. Phys. Rev. A 64, 061603 (2001)

    Article  ADS  Google Scholar 

  25. J.D. Tchinang Tchameu, C. Tchawoua, A.B. Togueu Motcheyo, Effects of next-nearest-neighbor interactions on discrete multibreathers corresponding to Davydov model with saturable nonlinearities. Phys. Lett. A 379, 2984 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. R.Y. Ondoua, J.C. Mimshe Fewu, D. Belobo Belobo, C.B. Tabi, H.P. Ekobena Fouda, Excitons dynamic in a three-stranded a-helix protein chains with diagonal and off-diagonal couplings: effects of strong long-range interactions. Eur. Phys. J. Plus 136, 274 (2021)

    Article  Google Scholar 

  27. H.C. Yuen, B.M. Lake, Instabilities of waves on deep water. Ann. Rev. Fluid Mech. 12, 303–334 (1980)

    Article  ADS  MATH  Google Scholar 

  28. P. Agrawal, Govind, Nonlinear Fiber Optics, 2nd edn. (Academic Press, San Diego (California), 1995)

    Google Scholar 

  29. E. Kenig, B.A. Malomed, M. Cross, R. Ifshitz, Intrinsic localized modes in parametrically driven arrays of nonlinear resonators. Phys. Rev. E 80, 046202 (2009)

    Article  ADS  Google Scholar 

  30. F. Palmero, J. Han, L.Q. English, T.J. Alexander, P.G. Kevrekidis, Multifrequency and edge breathers in the discrete sine-Gordon system via subharmonic driving: theory, computation and experiment. Phys. Lett. A 380, 402–407 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  31. L.M. Floria, J.J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model. Adv. Phy. 45, 505–598 (1996)

    Article  ADS  Google Scholar 

  32. J. Wu, R. Keolian, I. Rudnick, Observation of a nonpropagating hydrodynamic soliton. Phys. Rev. Lett. 52, 1421–1424 (1984)

    Article  ADS  Google Scholar 

  33. B. Denardo, W. Wright, S. Putterman, A. Larraza, Observation of a kink soliton on the surface of a liquid. Phys. Rev. Lett. 64, 1518–1521 (1990)

    Article  ADS  Google Scholar 

  34. X. Wang, R. Wei, Dynamics of Multisoliton Interactions in Parametrically Resonant Systems. Phys. Rev. Lett. 78, 2744–2747 (1997)

    Article  ADS  Google Scholar 

  35. O.M. Braun, Y. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods, and Applications (Springer-Verlag, Berlin, Heidelberg, 2004)

    Book  MATH  Google Scholar 

  36. C. Vasanthi, M. Latha, Heisenberg ferromagnetic spin chain with the bilinear and biquadratic interactions in (2 + 1) dimensions. Commun. Nonlinear Sci. Numer. Simulat. 28, 109–122 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. I. Barashenkov, M. Bogdan, V. Korobov, Stability diagram of the phase-locked solitons in the parametrically driven, damped nonlinear Schrodinger equation. Eur. Phys. Lett. 15, 113 (1991)

    Article  ADS  Google Scholar 

  38. Y. Xu, T.J. Alexander, H. Sidhu, P.G. Kevrekidis, Instability dynamics and breather formation in a horizontally shaken pendulum chain. Phys. Rev. E 90, 042921 (2014)

    Article  ADS  Google Scholar 

  39. V.M. Burlakov, Interference of mode instabilities and pattern formation in anharmonic lattices. Phys. Rev. Lett. 80, 3988 (1998)

    Article  ADS  Google Scholar 

  40. J. Leon, M. Manna, Discrete instability in nonlinear lattices. Phys. Rev. Lett. 83, 2324 (1999)

    Article  ADS  Google Scholar 

  41. A.V. Gorbach, M. Johanson, Gap and out-gap breathers in a binary modulated discrete nonlinear Schrödinger model. EPJ D 29, 77 (2004)

    ADS  Google Scholar 

  42. J. Meier, G.I. Stegeman, D.N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, J.S. Aitchison, Experimental observation of discrete modulational instability. Phys. Rev. Lett. 92, 163902 (2004)

    Article  ADS  Google Scholar 

  43. N.V. Alexeeva, I.V. Barashenkov, A.A. Sukhorukov, Y.S. Kivshar, Optical solitons in \(PT-symmetric\) nonlinear couplers with gain and loss. Phys. Rev. A 85, 063837 (2012)

    Article  ADS  Google Scholar 

  44. I.V. Barashenkov, S.V. Suchkov, A.A. Sukhorukov, S.V. Dmitriev, Y.S. Kivshar, Breathers in \(PT-symmetric\) optical couplers. Phys. Rev. A 86, 053809 (2012)

    Article  ADS  Google Scholar 

  45. N.V. Alexeeva, I.V. Barashenkov, Y.S. Kivshar, Solitons in \(PT-symmetric\) ladders of optical waveguides. New J. Phys. 19, 113032 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  46. E. Destyl, S.P. Nuiro, D.E. Pelinovsky, P. Poullet, Coupled pendula chains under parametric PT-symmetric driving force. Phys. Lett. A 381, 3884–3892 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. A. Chernyavsky, D.E. Pelinovsky, Breathers in Hamiltonian PT -symmetric chains of coupled pendula under a resonant periodic force. Symmetry 8(7), 59 (2016). https://doi.org/10.3390/sym8070059

    Article  ADS  MathSciNet  Google Scholar 

  48. A. Chernyavsky, D.E. Pelinovsky, Long-time stability of breathers in Hamiltonian PT-symmetric lattices. J. Phys. A: Math. Theor. 49, 475201 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. A. Kamdoum Kuitche, A.B. Togueu Motcheyo, T. Kanaa, C. Tchawoua, Supratransmission in transversely connected nonlinear pendulum pairs. Chaos Solitons Fractals 160, 112196 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  50. A. Jallouli, N. Kacem, N. Bouhaddi, Stabilization of solitons in coupled nonlinear pendulums with simultaneous external and parametric excitations. Commun. Nonlinear Sci. Numer. Simulat. 42, 1–11 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  51. P.G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numérical Computations and Physics Persective (Springer-Verlag, Berlin/Heidelberg, Germany, 2009)

    Book  Google Scholar 

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Acknowledgements

A. Kamdoum Kuitche would like to thank and express his sincere gratitude to A. S. Foualeng Kamga for the fruitful discussions. The authors acknowledge the Electronic Journal Delivery Service of the International Center of Theoretical Physics (ITCP) for providing valuable references used in this study. The authors wish to thank the anonymous reviewer and the editor for all their invaluable comments and criticisms. All of their suggestions were thoroughly followed, resulting in a substantial improvement of the present work. They also thank Prof Dmitry Pelinovsky for providing the model.

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Authors and Affiliations

  1. Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon

    A. Kamdoum Kuitche, A. B. Togueu Motcheyo & C. Tchawoua

  2. Department of Mechanical Engineering, Higher Technical Teacher’s Training College (ENSET) Ebolowa, University of Ebolowa, P.O. Box 886, Ebolowa, Cameroon

    A. B. Togueu Motcheyo & Thomas Kanaa

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  1. A. Kamdoum Kuitche
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  2. A. B. Togueu Motcheyo
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Correspondence to A. B. Togueu Motcheyo.

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Kamdoum Kuitche, A., Togueu Motcheyo, A.B., Kanaa, T. et al. Modulational instability in transversely connected nonlinear pendulum pairs. Eur. Phys. J. Plus 138, 142 (2023). https://doi.org/10.1140/epjp/s13360-023-03761-4

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