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

Peter J. Olver, Ari Stern

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)820-845
Number of pages26
JournalEuropean Journal of Applied Mathematics
Volume32
Issue number5
DOIs
StatePublished - Jan 12 2021

Bibliographical note

Funding Information:
The authors wish to thank Rajendra Beekie, Gong Chen, Burak Erdo?an, Natalie Sheils and Ferdinand Verhulst for helpful discussions and correspondence on FPUT and fractalisation. The authors also thank the referees for all their comments. Ari Stern was supported in part by a grant from the National Science Foundation (DMS-1913272).

Funding Information:
The authors wish to thank Rajendra Beekie, Gong Chen, Burak Erdog˘an, Natalie Sheils and Ferdinand Verhulst for helpful discussions and correspondence on FPUT and fractalisation. The authors also thank the referees for all their comments. Ari Stern was supported in part by a grant from the National Science Foundation (DMS-1913272).

Publisher Copyright:
© 2021 The Author(s). Published by Cambridge University Press.

Keywords

  • Fermi-Pasta-Ulam-Tsingou lattice
  • continuum model
  • dispersion
  • fractalisation
  • geometric integration
  • revival

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