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.

中文翻译：

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线性和非线性 Fermi-Pasta-Ulam-Tsingou 晶格中的色散分形

我们从解析和数值上研究周期性线性和非线性 Fermi-Pasta-Ulam-Tsingou 系统解的色散分形和量化。当受到周期性边界条件和不连续初始条件（例如阶跃函数）的影响时，FPUT 的线性化和非线性连续体模型都在非理性时间（由系数和区间长度确定）和量化分布（分段常数或其扰动）在合理的时间。我们有时在线性化 FPUT 链中观察到类似的效果吨这些模型有效的地方，即吨= O(H-2）， 在哪里H与质量间距成正比，或者等效地，质量数的倒数。对于非线性周期性 FPUT 系统，我们的数值结果表明存在小非线性时的行为有些相似，随着非线性力的幅度增加，这种行为会消失。然而，这些现象在很长的时间间隔内表现出来，随着质量数量的增加，对数值积分提出了严峻的挑战。即使使用此处使用的高阶分裂方法，我们的数值研究也仅限于非线性 FPUT 链，其质量数量少于明确解决该问题所需的质量。