Export file:

Format

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

Content

  • Citation Only
  • Citation and Abstract

Metastability of solitary waves in diatomic FPUT lattices

Department of Mathematics, Drexel University, 3141 Chestnut St, Philadelphia, PA 19104
$^\dagger$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

It is known that long waves in spatially periodic polymer Fermi-Pasta-Ulam-Tsingou lattices are well-approximated for long, but not infinite, times by suitably scaled solutions of Korteweg-de Vries equations. It is also known that dimer FPUT lattices possess nanopteron solutions, i.e., traveling wave solutions which are the superposition of a KdV-like solitary wave and a very small amplitude ripple. Such solutions have infinite mechanical energy. In this article we investigate numerically what happens over very long time scales (longer than the time of validity for the KdV approximation) to solutions of diatomic FPUT which are initially suitably scaled (finite energy) KdV solitary waves. That is we omit the ripple. What we find is that the solitary wave continuously leaves behind a very small amplitude “oscillatory wake.” This periodic tail saps energy from the solitary wave at a very slow (numerically sub-exponential) rate. We take this as evidence that the diatomic FPUT “solitary wave” is in fact quasi-stationary or metastable.
  Figure/Table
  Supplementary
  Article Metrics

Keywords diatomic Fermi-Pasta-Ulam-Tsingou lattices; metastability; Hamiltonian lattices; solitary waves; nonlinear waves

Citation: Nickolas Giardetti, Amy Shapiro, Stephen Windle, J. Douglas Wright. Metastability of solitary waves in diatomic FPUT lattices. Mathematics in Engineering, 2019, 1(3): 419-433. doi: 10.3934/mine.2019.3.419

References

  • 1. Faver TE, Wright JD (2018) Exact diatomic Fermi-Pasta-Ulam-Tsingou solitary waves with optical band ripples at infinity. SIAM J Math Anal 50: 182–250.    
  • 2. Lustri CJ, Porter MA (2018) Nanoptera in a period-2 Toda chain. SIAM J Appl Dyn Syst 17: 1182–1212.    
  • 3. Porter M, Daraio C, Szelengowicz I, et al. (2009) Highly nonlinear solitary waves in heterogeneous periodic granular media. Phys D 238: 666–676.    
  • 4. Gaison J, Moskow S, Wright JD, et al. (2014) Approximation of polyatomic FPU lattices by KdV equations. Multiscale Model Simul 12: 953–995.    
  • 5. Qin WX (2015) Wave propagation in diatomic lattices. SIAM J Math Anal 47: 477–497.    
  • 6. Betti M, Pelinovsky DE (2013) Periodic traveling waves in diatomic granular chains. J Nonlinear Sci 23: 689–730.    
  • 7. Chirilus-Bruckner M, Chong C, Prill O, et al. (2012) Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations. Discrete Contin Dyn Syst Ser S 5: 879–901. Available from: https://doi.org/10.3934/dcdss.2012.5.879.    
  • 8. Brillouin L (1953) Wave Propagation in Periodic Structures. Electric Filters and Crystal Lattices,2Eds., New York: Dover Publications, Inc.
  • 9. Tabata Y (1996) Stable solitary wave in diatomic Toda lattice. J Phys Soc Jpn 65: 3689–3691.    
  • 10. Okada Y, Watanabe S, Tanaca H (1990) Solitary wave in periodic nonlinear lattice. J Phys Soc Jpn 59: 2647–2658. Available from: https://doi.org/10.1143/JPSJ.59.2647.    
  • 11. Hoffman A, Wright JD (2017) Nanopteron solutions of diatomic Fermi-Pasta-Ulam-Tsingou lattices with small mass-ratio. Phys D 358: 33–59.    
  • 12. Vainchtein A, Starosvetsky Y, Wright JD, et al. (2016) Solitary waves in diatomic chains. Phys Rev E 93: 042210.    
  • 13. Schneider G, Wayne CE (1999) Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model, In: International Conference on Differential Equations,Vol. 1, 2 (Berlin, 1999), 390–404, World Sci. Publ., River Edge, NJ, 2000.
  • 14. Friesecke G, Pego RL (1999) Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12: 1601–1627.
  • 15. Friesecke G, Pego RL (2002) Solitary waves on FPU lattices: II. Linear implies nonlinear stability. Nonlinearity 15: 1343–1359.
  • 16. Friesecke G, Pego RL (2004) Solitary waves on Fermi-Pasta-Ulam lattices: III. Howland-type Floquet theory. Nonlinearity 17: 207–227.
  • 17. Friesecke G, Pego RL (2004) Solitary waves on Fermi-Pasta-Ulam lattices: IV. Proof of stability at low energy. Nonlinearity 17: 229–251.
  • 18. Mizumachi T (2009) Asymptotic stability of lattice solitons in the energy space. Commun Math Phys 288: 125–144.    
  • 19. Boyd JP (1998) Weakly Nonlocal Solitary Waves and Beyond-All-Orders Asymptotics Generalized Solitons and Hyperasymptotic Perturbation Theory , In Series: Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers, vol. 442. Available from: https://doi.org/10.1007/978-1-4615-5825-5.
  • 20. Faver T (2017) Nanopteron-stegoton traveling waves in spring dimer Fermi-Pasta-Ulam-Tsingou lattices, in press. Available from: https://arxiv.org/abs/1710.07376.
  • 21. Lombardi E (2000) Oscillatory Integrals and Phenomena Beyond All Algebraic Orders with Applications to Homoclinic Orbits in Reversible Systems, In series: Lecture Notes in Mathematics. Berlin: Springer-Verlag, vol. 1741. Available from: https://doi.org/10.1007/BFb0104102.
  • 22. Sun SM (1999) Non-existence of truly solitary waves in water with small surface tension. Proc Math Phys Eng Sci 455: 2191–2228.    
  • 23. Martínez AJ, Kevrekidis PG, Porter MA (2016) Superdiffusive transport and energy localization in disordered granular crystals. Phys Rev E 93: 022902. Available from: https://doi.org/10.1103/physreve.93.022902.    
  • 24. Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2Eds., In series: Springer Series in Computational Mathematics. Springer, Heidelberg, 2010, vol. 31.
  • 25. Beyn WJ, Thümmler V (2004) Freezing solutions of equivariant evolution equations. SIAM J Appl Dyn Syst 3: 85–116.    
  • 26. Beyn WJ, Otten D, Rottmann-Matthes J (2018) Computation and stability of traveling waves in second order evolution equations. SIAM J Numer Anal 56: 1786–1817.    
  • 27. Beale JT (1991) Exact solitary water waves with capillary ripples at infinity. Commun Pure Appl Math 44: 211–257.    
  • 28. Sun SM (1991) Existence of a generalized solitary wave solution for water with positive Bond number less than 1/3. J Math Anal Appl 156: 471–504.    
  • 29. LeVeque RJ, Yong DH (2003) Solitary waves in layered nonlinear media. SIAM J Appl Math 63: 1539–1560.    
  • 30. LeVeque RJ, Yong DH (2003) Phase plane behavior of solitary waves in nonlinear layered media, In: Hyperbolic Problems: Theory, Numerics, Applications. Berlin: Springer, 43–51.
  • 31. Kevrekidis PG, Stefanov AG, Xu H (2016) Traveling waves for the mass in mass model of granular chains. Lett Math Phys 106: 1067–1088.    
  • 32. Pnevmatikos S, Flytzanis N, Remoissenet M (1986) Soliton dynamics of nonlinear diatomic lattices. Phys Rev B 33: 2308–2321.    

 

This article has been cited by

  • 1. Christopher J. Lustri, Nanoptera and Stokes curves in the 2-periodic Fermi–Pasta–Ulam–Tsingou equation, Physica D: Nonlinear Phenomena, 2019, 132239, 10.1016/j.physd.2019.132239

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