Manevitch L I, Smirnov V V
Institute of Chemical Physics, RAS, 4 Kosygin Str., Moscow 119991, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Sep;82(3 Pt 2):036602. doi: 10.1103/PhysRevE.82.036602. Epub 2010 Sep 8.
We demonstrate that the modulation instability of the zone-boundary mode in a finite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficient condition for the efficient energy transfer by localized excitations. This transfer results from the exclusion of complete energy exchange between spatially different parts of the chain, and the excitation level corresponding to that turns out to be twice more than threshold of zone-boundary mode's instability. To obtain this result one needs in far going extension of the beating concept to a wide class of finite oscillatory chains. In turn, such an extension leads to description of energy exchange and transition to energy localization and transfer in terms of effective particles and limiting phase trajectories. The effective particles appear naturally when the frequency spectrum crowding ensures the resonance interaction between zone-boundary and two nearby nonlinear normal modes, but there are no additional resonances. We show that the limiting phase trajectories corresponding to the most intensive energy exchange between effective particles can be considered as an alternative to nonlinear normal modes, which describe the stationary process.
我们证明，在有限（周期）费米 - 帕斯塔 - 乌拉姆链中，区域边界模式的调制不稳定性是通过局域激发实现高效能量转移的必要但非充分条件。这种转移源于链的空间不同部分之间完全能量交换的排除，并且与之对应的激发水平结果是区域边界模式不稳定性阈值的两倍多。为了获得这个结果，需要将拍频概念大幅扩展到一类广泛的有限振荡链。反过来，这种扩展导致了根据有效粒子和极限相位轨迹来描述能量交换以及向能量局域化和转移的转变。当频谱拥挤确保区域边界与两个相邻非线性正常模式之间的共振相互作用，但不存在额外共振时，有效粒子自然出现。我们表明，对应于有效粒子之间最强烈能量交换的极限相位轨迹可以被视为描述平稳过程的非线性正常模式的替代物。
