Localized waves in nonlinear oscillator chains.
Localized waves in nonlinear oscillator chains.
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DOI：
10.1063/1.1836151
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发表时间：
2005-03
期刊：
影响因子：
2.9
通讯作者：
G. Iooss;G. James
中科院分区：
文献类型：
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作者：
G. Iooss;G. James
本文回顾了耦合振子非线性链中空间局域波存在性的结果，并为费米-帕斯塔-乌拉姆（FPU）晶格提供了新的结果。局部解决方案包括永久形式的孤立波和移动呼吸器，它们在以恒定速度移动的参考系统中出现时间周期性。对于 FPU 晶格，我们分析呼吸周期和逆速度相称的情况。我们采用 Iooss 和 Kirchgassner 在行波情况下引入的中心流形简化方法，将问题局部简化为有限维可逆微分方程。简化系统的主要部分是可积的，并且如果满足相互作用势的硬化条件，则允许解与准周期轨道同宿。这些轨道对应于叠加在准周期振荡尾部上的近似移动呼吸器解。在一般情况下，它们对整个系统的持久性问题仍然存在。如果呼吸周期等于反速度的两倍，我们会解决这个问题的偶势，并证明在这种情况下存在叠加在指数小周期性尾部上的精确行进呼吸解。
This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.