Dispersive Fractalization in Linear and Nonlinear Fermi-Pasta-Ulam-Tsingou Lattices

TL;DR

This paper explores dispersive fractalization and quantization phenomena in linear and nonlinear Fermi-Pasta-Ulam-Tsingou lattices, revealing fractal and quantized solution profiles at specific times and discussing the challenges in numerical analysis of these effects.

Contribution

It provides analytical and numerical evidence of fractalization and quantization in FPUT systems, highlighting the effects of nonlinearity and the difficulties in long-term numerical simulations.

Findings

Fractal solution profiles occur at irrational times in linear models.

Quantized, piecewise constant profiles appear at rational times.

Nonlinear effects diminish with increasing nonlinearity and are observed over long times.

Abstract

We investigate, both analytically and numerically, dispersive fractalization and quantization 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 linearized 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 quantized profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearized FPUT chain at times $t$ where these models have validity, namely $t = \mathrm{O}(h^{-2})$, 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 behavior 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.