任意非线性声学链中的q呼吸子和热化
q-Breathers and thermalization in acoustic chains with arbitrary nonlinearity index
摘要：
非线性形状晶格动力学影响振动光谱，运输和热化现象。除了呼吸子和孤子，人们发现第三类基本的非线性模式-- $q$ -呼吸子--非线性晶格中的周期轨道，指数地局域在倒易模式空间中。迄今为止，对$q$ -呼吸子的研究仅限于相互作用势中的三次和四次非线性。本文研究了声学链中任意非线性指数$γ$的情况。我们发现了$q$ -呼吸子的离域和稳定性阈值随系统大小的变化规律：存在一个临界指数$γ^*=6$，低于该指数时，两个阈值（非线性强度）趋于零，高于该指数时，两个阈值发散.我们还表明，这个临界指数值是决定性的热化的存在或缺乏。对于一般的相互作用势，模空间局域动力学仅由幂级数展开中的三个最低阶非线性项决定。
英文摘要：
Nonlinearity shapes lattice dynamics affecting vibrational spectrum, transport and thermalization phenomena. Beside breathers and solitons one finds the third fundamental class of nonlinear modes -- $q$-breathers -- periodic orbits in nonlinear lattices, exponentially localized in the reciprocal mode space. To date, the studies of $q$-breathers have been confined to the cubic and quartic nonlinearity in the interaction potential. In this paper we study the case of arbitrary nonlinearity index $γ$ in an acoustic chain. We uncover qualitative difference in the scaling of delocalization and stability thresholds of $q$-breathers with the system size: there exists a critical index $γ^*=6$, below which both thresholds (in nonlinearity strength) tend to zero, and diverge when above. We also demonstrate that this critical index value is decisive for the presence or absense of thermalization. For a generic interaction potential the mode space localized dynamics is determined only by the three lowest order nonlinear terms in the power series expansion.
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