一类弱耦合振子Hamilton网络呼吸子的渐近稳定性
Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators
摘要：
考虑弱耦合非谐振子的哈密顿链。众所周知，如果耦合足够弱，则系统允许在空间中指数局部化的周期解族（呼吸子）。本文证明了这类解在能量空间中的渐近稳定性。证明是基于两个步骤：第一，我们使用正则扰动理论，把系统在一个适当的正规形在附近的呼吸，第二，我们使用色散，以证明渐近稳定性。结果的主要限制在于，现场电位的非线性部分需要在原点有一个8阶的零点。从技术的角度来看，该理论不同于哈密顿偏微分方程，由于事实上，呼吸是不是一个相对平衡的系统。
英文摘要：
We consider a Hamiltonian chain of weakly coupled anharmonic oscillators. It is well known that if the coupling is weak enough then the system admits families of periodic solutions exponentially localized in space (breathers). In this paper we prove asymptotic stability in energy space of such solutions. The proof is based on two steps: first we use canonical perturbation theory to put the system in a suitable normal form in a neighborhood of the breather, second we use dispersion in order to prove asymptotic stability. The main limitation of the result rests in the fact that the nonlinear part of the on site potential is required to have a zero of order 8 at the origin. From a technical point of view the theory differs from that developed for Hamiltonian PDEs due to the fact that the breather is not a relative equilibrium of the system.
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