标题： 可积的离散波近似方法的颗粒链分散冲击波 显示英文标题

标题： Integrable Approximations of Dispersive Shock Waves of the Granular Chain

摘要： 在本工作中，我们重新研究了具有预压缩的颗粒链中的激波动力学。通过将模型近似为$\alpha$-Fermi-Pasta-Ulam-Tsingou 链，我们利用后者在应变变量公式中与两个独立可积模型的联系，一个是连续的，即 KdV 方程，另一个是离散的，即 Toda 晶格。我们运用 Whitham 调制理论对这些可积系统的分析以及它们色散激波的解析近似，通过到颗粒晶体的约化连接，提供对颗粒问题激波的近似。原始颗粒链及其基于可积系统的色散激波的详细数值比较在广泛的参数范围内非常有利。随着解的振幅参数增加，(近似)理论与数值计算之间的渐进偏差被量化并进行了讨论。 显示英文摘要

摘要： In the present work we revisit the shock wave dynamics in a granular chain with precompression. By approximating the model by an $\alpha$-Fermi-Pasta-Ulam-Tsingou chain, we leverage the connection of the latter in the strain variable formulation to two separate integrable models, one continuum, namely the KdV equation, and one discrete, namely the Toda lattice. We bring to bear the Whitham modulation theory analysis of such integrable systems and the analytical approximation of their dispersive shock waves in order to provide, through the lens of the reductive connection to the granular crystal, an approximation to the shock wave of the granular problem. A detailed numerical comparison of the original granular chain and its approximate integrable-system based dispersive shocks proves very favorable in a wide parametric range. The gradual deviations between (approximate) theory and numerical computation, as amplitude parameters of the solution increase are quantified and discussed.