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

• Published in: European journal of applied mathematics Vol. 32; no. 5; pp. 820 - 845
• Main Authors: OLVER, PETER J., STERN, ARI
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.10.2021
• Subjects: Applied mathematics; Boundary conditions; Chains; Computers; Continuum modeling; Dispersion; Fractal models; Fractals; Initial conditions; Investigations; Lattices (mathematics); Linearization; Nonlinearity; Numerical analysis; Numerical integration; Numerical methods; Partial differential equations; Perturbation; Step functions; 65P10; 28A80; 37K60; 35Q53; 70H08; revival; continuum model; geometric integration; Fermi–Pasta–Ulam–Tsingou lattice; 70F45; 42A32; dispersion; fractalisation
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S095679252000042X

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.