Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T08:30:48.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Dispersive fractalisation in linear and nonlinear Fermi–Pasta–Ulam–Tsingou lattices

Part of: Hamiltonian and Lagrangian mechanics Numerical problems in dynamical systems Infinite-dimensional Hamiltonian systems Harmonic analysis in one variable Equations of mathematical physics and other areas of application Classical measure theory

Published online by Cambridge University Press:  12 January 2021

PETER J. OLVER Open the ORCID record for PETER J. OLVER [Opens in a new window]  and
ARI STERN
PETER J. OLVER
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA email: [email protected]
ARI STERN
Affiliation:
Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO 63130, USA email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at times t where these models have validity, namely t = O(h−2), where h is proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.

Keywords

Fermi–Pasta–Ulam–Tsingou latticedispersionfractalisationrevivalcontinuum modelgeometric integration

MSC classification

Primary: 37K60: Lattice dynamics
Secondary: 28A80: Fractals 35Q53: KdV-like equations (Korteweg-de Vries) 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) 65P10: Hamiltonian systems including symplectic integrators 70F45: Infinite particle systems 70H08: Nearly integrable Hamiltonian systems, KAM theory
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 820 - 845
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

References

Arioli, G., Koch, H. & Terracini, S. (2005) Two novel methods and multi-mode periodic solutions for the Fermi-Pasta-Ulam model. Commun. Math. Phys. 255, 1–19.CrossRefGoogle Scholar
Benjamin, T. B., Bona, J. L. & Mahony, J. J. (1972) Model equations for long waves in nonlinear dispersive systems. Phil. Trans. Roy. Soc. London A 272, 47–78.Google Scholar
Berry, M. V. (1996) Quantum fractals in boxes. J. Phys. A 29, 6617–6629.CrossRefGoogle Scholar
Berry, M. V. & Klein, S. (1996) Integer, fractional and fractal Talbot effects. J. Mod. Opt. 43, 2139–2164.CrossRefGoogle Scholar
Berry, M. V., Marzoli, I. & Schleich, W. (2001) Quantum carpets, carpets of light. Phys. World 14 (6), 39–44.CrossRefGoogle Scholar
Blanes, S. & Moan, P. C. (2002) Practical symplectic partitioned Runge–Kutta and Runge–Kutta–Nyström methods. J. Comput. Appl. Math. 142, 313–330.CrossRefGoogle Scholar
Bona, J. L. & Saut, J.-C. (1993) Dispersive blowup of solutions of generalized Korteweg–de Vries equations. J. Diff. Eq. 103, 3–57.CrossRefGoogle Scholar
Boulton, L., Olver, P. J., Pelloni, B. & Smith, D. A. (2020) New revival phenomena for linear integro-differential equations. arXiv:2010.01320.Google Scholar
Boussinesq, J. (1877) Essai sur la théorie des eaux courantes. Mém. Acad. Sci. Inst. Nat. France 23 (1), 1–680.Google Scholar
Bruggeman, R. & Verhulst, F. (2019) Near-Integrability and recurrence in FPU chains with alternating masses. J. Nonlinear Sci. 29, 183–206.CrossRefGoogle Scholar
Calogero, F. (1971) Solution of the one-dimensional n-body problems with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436.CrossRefGoogle Scholar
Chen, G. & Olver, P. J. (2013) Dispersion of discontinuous periodic waves. Proc. Roy. Soc. London A 469, 20120407.Google Scholar
Chousionis, V., Erdoğan, M. B. & Tzirakis, N. (2015) Fractal solutions of linear and nonlinear dispersive partial differential equations. Proc. London Math. Soc. 110, 543–564.Google Scholar
Daripa, P. & Hua, W. (1999) A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: filtering and regularization techniques. Appl. Math. Comput. 101, 159–207.Google Scholar
Dauxois, T. (2008) Fermi, Pasta, Ulam, and a mysterious lady. Phys. Today 61(1), 55–57.CrossRefGoogle Scholar
Dauxois, T., Peyrard, M. & Ruffo, S. (2005) The Fermi-Pasta-Ulam ‘numerical experiment’: history and pedagogical perspectives. Eur. J. Phys. 26, S3–S11.CrossRefGoogle Scholar
Drazin, P. G. & Johnson, R. S. (1989) Solitons: An Introduction, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Dubrovin, B. A., Matveev, V. B. & Novikov, S. P. (1976) Nonlinear equations of Korteweg-De Vries type, finite-zone linear operators and Abelian varieties. Russian Math. Surv. 31, 56–134.CrossRefGoogle Scholar
Erdoğan, M. B. & Shakan, G. (2019) Fractal solutions of dispersive partial differential equations on the torus. Selecta Math. 25, 11.CrossRefGoogle Scholar
Erdoğan, M. B. & Tzirakis, N. (2013) Talbot effect for the cubic nonlinear Schrödinger equation on the torus. Math. Res. Lett. 20, 1081–1090CrossRefGoogle Scholar
Erdoğan, M. B. & Tzirakis, N. (2016) Dispersive Partial Differential Equations: Wellposedness and Applications. London Mathematical Society Student Texts, vol. 86, Cambridge University Press, Cambridge.Google Scholar
Fermi, E., Pasta, J. & Ulam, S. (1974) Studies of nonlinear problems. I., Los Alamos Report LA1940, 1955. In: Newell, A. C. (editor), Nonlinear Wave Motion, Lectures in Applied Mathematics, vol. 15, American Mathematical Society, Providence, R.I., pp. 143–156.Google Scholar
Ford, J. (1992) The Fermi-Pasta-Ulam problem: Paradox turns discovery. Phys. Rep. 213, 271–310.CrossRefGoogle Scholar
Friesecke, G. & Wattis, J. (1994) Existence theorem for solitary waves on lattices. Commun. Math. Phys. 161, 391–418.CrossRefGoogle Scholar
Galgani, L., Giorgilli, A., Martinoli, A. & Vanzini, S. (1992) On the problem of energy partition for large systems of the Fermi–Pasta–Ulam type: analytical and numerical estimates. Physica D 59, 334–348.CrossRefGoogle Scholar
Hairer, E., Lubich, C. & Wanner, G. (2006) Geometric Numerical Integration, 2nd ed., Springer–Verlag, New York.Google Scholar
Hofmanová, M. & Schratz, K. (2017) An exponential-type integrator for the KdV equation. Numer. Math. 136, 1117–1137.CrossRefGoogle Scholar
Lennard–Jones, J. E. (1972) On the determination of molecular fields. Phys. Rev. A 5, 1372–1376.Google Scholar
Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M. & Vulpiani, A. (1983) Relaxation to different stationary states in the Fermi-Pasta-Ulam model. Phys. Rev. A 28, 3544–3552.CrossRefGoogle Scholar
Livi, R., Pettini, M., Ruffo, S., Sparpaglione, M. & Vulpiani, A. (1985) Equipartition threshold in nonlinear large Hamiltonian systems: the Fermi–Pasta–Ulam model. Phys. Rev. A 31, 1039–1045.CrossRefGoogle ScholarPubMed
McKean, H. P. & van Moerbeke, P. (1975) The spectrum of Hill’s equation. Invent. Math. 30, 217–274.CrossRefGoogle Scholar
McLachlan, R. I. & Quispel, G. R. W. (2002) Splitting methods. Acta Numer. 11, 341–434.CrossRefGoogle Scholar
McLachlan, R. I. & Stern, A. (2014) Modified trigonometric integrators. SIAM J. Numer. Anal. 52, 1378–1397.CrossRefGoogle Scholar
Moser, J. (1975) Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math. 16, 197–220.CrossRefGoogle Scholar
Olver, P. J. (1984) Hamiltonian perturbation theory and water waves. Contemp. Math. 28, 231–249.CrossRefGoogle Scholar
Olver, P. J. (1984) Hamiltonian and non–Hamiltonian models for water waves. In: Ciarlet, P. G. and Roseau, M. (editors), Trends and Applications of Pure Mathematics to Mechanics, Lecture Notes in Physics, vol. 195, Springer–Verlag, New York, pp. 273–290.CrossRefGoogle Scholar
Olver, P. J. (2010) Dispersive quantization. Amer. Math. Monthly 117, 599–610.CrossRefGoogle Scholar
Olver, P. J. (2014) Introduction to Partial Differential Equations, Undergraduate Texts in Mathematics, Springer, New York.Google Scholar
Olver, P. J. & Shakiban, C. (2018) Applied Linear Algebra, 2nd ed., Undergraduate Texts in Mathematics, Springer, New York.Google Scholar
Olver, P. J. & Tsatis, E. (2018) Points of constancy of the periodic linearized Korteweg–deVries equation. Proc. Roy. Soc. London A 474, 20180160.Google Scholar
Oskolkov, K. I. (1992) A class of I.M. Vinogradov’s series and its applications in harmonic analysis. In: Progress in Approximation Theory, Springer Series in Computational Mathematics, vol. 19, Springer, New York, pp. 353–402.CrossRefGoogle Scholar
Pankov, A. (2005) Travelling Waves and Periodic Oscillations in Fermi–Pasta–Ulam Lattices, Imperial College Press, London.CrossRefGoogle Scholar
Porter, M. A., Zabusky, N. J., Hu, B. & Campbell, D. K. (2009) Fermi, Pasta, Ulam, and the birth of experimental mathematics. Am. Sci. 97(3), 214–221.CrossRefGoogle Scholar
Pöschel, J. (2001) A lecture on the classical KAM theorem. Proc. Sympos. Pure Math. 69, 707–732.Google Scholar
Rink Rink, B. (2001) Symmetry and resonance in periodic FPU chains. Commun. Math. Phys. 218, 665–685.CrossRefGoogle Scholar
Rodnianski, I. (2000) Fractal solutions of the Schrödinger equation. Contemp. Math. 255, 181–187.CrossRefGoogle Scholar
Rosenau, P. (1986) Dynamics of nonlinear mass-spring chains near the continuum limit. Phys. Lett. A 118, 222–227.CrossRefGoogle Scholar
Rosenau, P. (1987) Dynamics of dense lattices. Phys. Rev. B 36, 5868–5876.CrossRefGoogle ScholarPubMed
Scott, A. C. (1985) Soliton oscillations in DNA. Phys. Rev. A 31, 3518–3519.CrossRefGoogle ScholarPubMed
Smoller, J. (1994) Shock Waves and Reaction–Diffusion Equations, 2nd ed., Springer–Verlag, New York.CrossRefGoogle Scholar
Stern, A. & Grinspun, E. (2009) Implicit-Explicit variational integration of highly oscillatory problems. Multiscale Model. Simul. 7, 1779–1794.CrossRefGoogle Scholar
Strang, G. (1968) On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517.CrossRefGoogle Scholar
Sutherland, B. (1972) Exact results for a quantum many-body problem in one-dimension. II. Phys. Rev. A 5, 1372–1376.CrossRefGoogle Scholar
Talbot, H. F. (1836) Facts related to optical science. No. IV. Philos. Mag. 9, 401–407.Google Scholar
Toda, M. (1981) Theory of Nonlinear Lattices, Springer–Verlag, New York.CrossRefGoogle Scholar
Tuck, J. L. & Menzel, M. T. (1972) The superperiod of the nonlinear weighted string (FPU) problem. Adv. Math. 9, 399–407.CrossRefGoogle Scholar
Ulam, S. M. (1976) Adventures of a Mathematician, Scribner, New York.CrossRefGoogle Scholar
Vrakking, M. J. J., Villeneuve, D. M. & Stolow, A. (1996) Observation of fractional revivals of a molecular wavepacket. Phys. Rev. A 54, R37–40.CrossRefGoogle Scholar
Weissert, T. P. (1997) The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem, Springer, New York.CrossRefGoogle Scholar
Whitham, G. B. (1974) Linear and Nonlinear Waves, John Wiley & Sons, New York.Google Scholar
Yeazell, J. A. & Stroud, C. R., Jr. (1991) Observation of fractional revivals in the evolution of a Rydberg atomic wave packet. Phys. Rev. A 43, 5153–5156.CrossRefGoogle ScholarPubMed
Zabusky, N. J. (1981) Computational synergetics and mathematical innovation. J. Comp. Phys. 43, 195–249.CrossRefGoogle Scholar
Zabusky, N. J. & Deem, G. S. (1967) Dynamics of nonlinear lattices I. Localized optical excitations, acoustic radiation, and strong nonlinear behavior. J. Comp. Phys. 2, 126–153.Google Scholar
Zabusky, N. J. & Kruskal, M. D. (1965) Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243.CrossRefGoogle Scholar