论文标题
线性和非线性Fermi-Pasta-Pasta-ulam-tsingou lattices中的色散分形化
Dispersive Fractalization in Linear and Nonlinear Fermi-Pasta-Ulam-Tsingou Lattices
论文作者
论文摘要
我们在分析和数值上研究了分散分化和对周期性线性和非线性Fermi-Pasta-ulam-tsingou系统的解决方案的量化。当受到周期性边界条件和不连续的初始条件的约束时，例如，一个阶跃函数（用于FPUT的线性化和非线性连续模型）在不合理的时间（由于系数和间隔的长度确定）和量化的曲线（在有理时间的零件常数或触及范围内）显示出分形溶液剖面。我们在线性化的fput链中观察到类似的效果，有时这些模型具有有效性，即$ t = \ mathrm {o}（h^{ - 2}）$，其中$ h $与质量间间距成正比或等效地，质量的数量。对于非线性周期性FPUT系统，我们的数值结果表明在存在小非线性的情况下的行为有些相似，随着非线性力量的增加而消失。但是，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而对数值整合构成了严重的挑战。即使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这个问题所需的质量。
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.
