Starosvetsky Yuli, Vakakis Alexander F
Department of Mechanical Science and Engineering, University of Illinois, Urbana-Champaign, Urbana, Illinois 61822, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Aug;82(2 Pt 2):026603. doi: 10.1103/PhysRevE.82.026603. Epub 2010 Aug 20.
We study a class of strongly nonlinear traveling waves and localized modes in one-dimensional homogeneous granular chains with no precompression. Until now the only traveling-wave solutions known for this class of systems were the single-hump solitary waves studied by Nesterenko in the continuum approximation limit. Instead, we directly study the discrete strongly nonlinear governing equations of motion of these media without resorting to continuum approximations or homogenization, which enables us to compute families of stable multihump traveling-wave solutions with arbitrary wavelengths. We develop systematic semianalytical approaches for computing different families of nonlinear traveling waves parametrized by spatial periodicity (wave number) and energy, and show that in a certain asymptotic limit, these wave families converge to the known single-hump solitary wave studied by Nesterenko. In addition, we demonstrate the existence of an additional class of stable strongly localized out-of-phase standing waves in perfectly homogeneous granular chains with no precompression or disorder. Until now such localized solutions were known to exist only in granular chains with strong precompression. Our findings indicate that homogeneous granular chains possess complex intrinsic nonlinear dynamics, including intrinsic nonlinear energy transfer and localization phenomena.
我们研究一类无预压缩的一维均匀颗粒链中的强非线性行波和局域模。到目前为止，这类系统已知的唯一行波解是涅斯捷连科在连续近似极限下研究的单峰孤立波。相反，我们直接研究这些介质的离散强非线性运动控制方程，而不借助连续近似或均匀化，这使我们能够计算具有任意波长的稳定多峰行波解族。我们开发了系统的半解析方法来计算由空间周期性（波数）和能量参数化的不同非线性行波族，并表明在一定的渐近极限下，这些波族收敛到涅斯捷连科研究的已知单峰孤立波。此外，我们证明了在没有预压缩或无序的完全均匀颗粒链中存在另一类稳定的强局域异相驻波。到目前为止，已知这种局域解仅存在于具有强预压缩的颗粒链中。我们的研究结果表明，均匀颗粒链具有复杂的内在非线性动力学，包括内在非线性能量转移和局域化现象。
