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

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

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: olver@umn.edu
ARI STERN
Affiliation:
Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, MO 63130, USA email: stern@wustl.edu
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Abstract

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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
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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

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