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

Peter J Olver,Ari Stern

doi:10.1017/s095679252000042x

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 timestwhere these models have validity, namelyt= O(h−2), wherehis 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.

Highlights

Introduction and historical perspectiveThe early 1950s witnessed the birth of the world’s first all purpose electronic computers,1 thereby bringing hitherto infeasible numerical calculations into the realm of possibility
The surprise was that the Fermi–Pasta– Ulam–Tsingou2 (FPUT) dynamics, at least on moderately long time intervals, did not proceed to thermalisation, i.e., exhibit ergodicity, as expected, but rather exhibited an unanticipated recurrence, in which energy from the low frequency modes would initially spread out into some of the higher modes but, after a certain time period, the system would almost entirely return to its initial configuration, as eloquently described in the above quote from Ulam’s autobiography
The subsequent rediscovery of this remarkable phenomenon by the first author, in the context of the periodic linearised Korteweg–deVries equation, [37, 38], showed that fractalisation and quantisation phenomena appear in a wide range of linear dispersivedifferential equations, including models arising in fluid mechanics, plasma dynamics, elasticity, DNA dynamics, and elsewhere

Summary

Introduction and historical perspective

The early 1950s witnessed the birth of the world’s first all purpose electronic computers, thereby bringing hitherto infeasible numerical calculations into the realm of possibility. The subsequent rediscovery of this remarkable phenomenon by the first author, in the context of the periodic linearised Korteweg–deVries equation, [37, 38], showed that fractalisation and quantisation phenomena appear in a wide range of linear dispersive (integro-)differential equations, including models arising in fluid mechanics, plasma dynamics, elasticity, DNA dynamics, and elsewhere Such linear systems exhibit a fascinating range of as yet poorly understood dynamical behaviours, whose qualitative features are tied to the large wave number asymptotics of the underlying dispersion relation. When the system is subjected to a highly concentrated initial displacement – displacing a single mass in the FPUT system or imposing a delta function in the Korteweg–deVries equation – at rational long range times the Korteweg–deVries profiles exhibit revival by reconcentrating at a finite number of locations, whereas the linear FPUT profile remains in a similar fractal form as its nearby irrational times. We defer the further development of more powerful analytical and numerical tools in order to conclusively deal with this intriguing problem

Fermi–Pasta–Ulam–Tsingou chains and their continuum models

The Riemann problem

Numerical investigation of nonlinear FPUT chains

Discussion