Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Energy transmission in Hamiltonian systems of globally interacting particles with Klein-Gordon on-site potentials

1 Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, Ags. 20131, Mexico
2 Department of Mathematics, Nazarbayev University, Kabanbay-Batyr 53, 010000 Astana, Republic of Kazakhstan
3 Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4 Athens, GR-11527, Greece
$^\dagger$This contribution is part of the Special Issue: Hamiltonian Lattice Dynamics
  Guest Editors: Simone Paleari; Tiziano Penati
  Link: http://www.aimspress.com/newsinfo/1165.html

Special Issues: Hamiltonian Lattice Dynamics

We consider a family of 1-dimensional Hamiltonian systems consisting of a large number of particles with on-site potentials and global (long range) interactions. The particles are initially at rest at the equilibrium position, and are perturbed sinusoidally at one end using Dirichlet data, while at the other end we place an absorbing boundary to simulate a semi-infinite medium. Using such a lattice with quadratic particle interactions and Klein-Gordon type on-site potential, we use a parameter $0\leq\alpha<\infty$ as a measure of the ``length'' of interactions, and show that there is a sharp threshold above which energy is transmitted in the form of large amplitude nonlinear modes, as long as driving frequencies $\Omega$ lie in the forbidden band-gap of the system. This process is called nonlinear supratransmission and is investigated here numerically to show that it occurs at higher amplitudes the longer the range of interactions, reaching a maximum at a value $\alpha=\alpha_{max} \lesssim 1.5$ that depends on $\Omega$. Below this $\alpha_{max}$ supratransmission thresholds decrease sharply to values lower than the nearest neighbor $\alpha=\infty$ limit. We give a plausible argument for this phenomenon and conjecture that similar results are present in related systems such as the sine-Gordon, the nonlinear Klein-Gordon and the double sine-Gordon type.
  Figure/Table
  Supplementary
  Article Metrics

Keywords nonlinear supratransmission; globally interacting systems; on-site potentials

Citation: Jorge E. Macías-Díaz, Anastasios Bountis, Helen Christodoulidi. Energy transmission in Hamiltonian systems of globally interacting particles with Klein-Gordon on-site potentials. Mathematics in Engineering, 2019, 1(2): 343-358. doi: 10.3934/mine.2019.2.343

References

  • 1. Remoissenet M (2013) Waves Called Solitons: Concepts and Experiments, 3 Eds., New York: Springer Science & Business Media.
  • 2. Lomdahl PS, Soerensen OH, Christiansen PL (1982) Soliton excitations in Josephson tunnel junctions. Phys Rev B 25: 5737.    
  • 3. Makhankov VG, Bishop AR, Holm DD (1995) Nonlinear Evolution Equations and Dynamical Systems Needs' 94. New York: World Scientific.
  • 4. Geniet F, Leon J (2002) Energy transmission in the forbidden band gap of a nonlinear chain. Phys Rev Lett 89: 134102.    
  • 5. Chevriaux D, Khomeriki R, Leon J (2006) Theory of a Josephson junction parallel array detector sensitive to very weak signals. Phys Rev B 73: 214516.    
  • 6. Geniet F, Leon J (2003) Nonlinear supratransmission. J Phys: Condens Matter 15: 2933–2949.    
  • 7. Khomeriki R, Lepri S, Ruffo S (2004) Nonlinear supratransmission and bistability in the Fermi- Pasta-Ulam model. Phys Rev E 70: 066626.    
  • 8. Leon J, Spire A (2004) Gap soliton formation by nonlinear supratransmission in Bragg media. Phys Lett A 327: 474–480.    
  • 9. Khomeriki R, Leon J (2005) Bistability in the sine-Gordon equation: The ideal switch. Phys Rev E 71: 056620.    
  • 10. Macías-Díaz JE, Medina-Ramírez I (2009) Nonlinear supratransmission and nonlinear bistability in a forced linear array of anharmonic oscillators: A computational study. Int J Mod Phys C 20: 1911–1923.    
  • 11. Macías-Díaz JE, Ruiz-Ramírez J, Flores-Oropeza LA ( 2009) Computational study of the transmission of energy in a two-dimensional lattice with nearest-neighbor interactions. Int J Mod Phys C 20: 1933–1943.
  • 12. Bodo B, Morfu S, Marquié P, et al. (2010) Klein-Gordon electronic network exhibiting supratransmission effect. Electron Lett 46: 123–124.    
  • 13. Tchinang Tchameu JD, Tchawoua C, Togueu Motcheyo AB (2016) Nonlinear supratransmission of multibreathers in discrete nonlinear Schrödinger equation with saturable nonlinearities. Wave Motion 65: 112–118.    
  • 14. Togueu Motcheyo AB, Tchinang Tchameu JD, Siewe Siewe M, et al. (2017) Homoclinic nonlinear band gap transmission threshold in discrete optical waveguide arrays. Commun Nonlinear Sci 50: 29–34.    
  • 15. Togueu Motcheyo AB, Tchinang Tchameu JD, Fewo SI, et al. (2017) Chameleon's behavior of modulable nonlinear electrical transmission line. Commun Nonlinear Sci 53: 22–30.    
  • 16. Coronel-Escamilla A, Gómez-Aguilar JF, Alvarado-Méndez E, et al. (2016) Fractional dynamics of charged particles in magnetic fields. Int J Mod Phys C 27: 1650084.    
  • 17. Benetti FPC, Ribeiro-Teixeira AC, Pakter R, et al. (2014) Nonequilibrium stationary states of 3D self-gravitating systems. Phys Rev Lett 113: 100602.    
  • 18. Miloshevich G, Nguenang JP, Dauxois T, et al. (2017) Traveling solitons in long-range oscillator chains. J Phys A: Math Theor 50: 12LT02.
  • 19. Campa A, Dauxois T, Ruffo S (2009) Statistical mechanics and dynamics of solvable models with long-range interactions. Phys Rep 480: 57–159.    
  • 20. Miele A, Dekker J (2008) Long-range chromosomal interactions and gene regulation. Mol Biosyst 4: 1046–1057.    
  • 21. Christodoulidi H, Tsallis C, Bountis T (2014) Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics. Europhys Lett 108: 40006.    
  • 22. Christodoulidi H, Bountis T, Tsallis C, et al. (2016) Dynamics and statistics of the Fermi-Pasta- Ulam $\beta$-model with different ranges of particle interactions. J Stat Mech: Theory Exp 12: 123206.
  • 23. Macías-Díaz JE, Bountis A (2018) Supratransmission in $\beta$-Fermi-Pasta-Ulam chains with different ranges of interactions. Commun Nonlinear Sci 63: 307–321.    
  • 24. Tarasov VE (2006) Continuous limit of discrete systems with long-range interaction. J Phys A: Math Gen 39: 14895–14910.    
  • 25. Macías-Díaz JE (2018) An explicit dissipation-preserving method for Riesz space-fractional nonlinear wave equations in multiple dimensions. Commun Nonlinear Sci 59: 67–87.    
  • 26. Chendjou GNB, Nguenang JP, Trombettoni A, et al. (2018) Fermi-Pasta-Ulam chains with harmonic and anharmonic long-range interactions. Communi Nonlinear Sci 60: 115–127.    
  • 27. Macías-Díaz JE, Puri A (2008) An energy-based computational method in the analysis of the transmission of energy in a chain of coupled oscillators. J Comput Appl Math 214: 393–405.    
  • 28. Macías-Díaz JE (2010) On the simulation of the energy transmission in the forbidden band-gap of a spatially discrete double sine-Gordon system. Comput Phys Commun 181: 1842–1849.    
  • 29. Christodoulidi H, Bountis T, Drossos L (2018) The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential. Eur Phys J Spec Top 227: 563–573.    

 

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved