Relaxation to Intermediate Attractors in Nonlinear Wave Equations
Relaxation to Intermediate Attractors in Nonlinear Wave Equations
复制标题
非线性波动方程中中间吸引子的弛豫
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
Nikodem Szpak
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文献类型：
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作者：
Nikodem Szpak
我们研究了奇点形成阈值附近非线性波动方程的动力学，这是由在临界引力坍缩中观察到的一些意想不到的特征所驱动的。其中一个特征，即在奇点形成的阈值处动力学中存在一个通用的中间吸引子，被更详细地研究了。结果是演化的方案：对于所有足够接近阈值的初始数据，解达到一个通用静态解，它起中间吸引子的作用。解在吸引子附近停留一段时间，吸引子以一种简单的方式与初始数据的参数进行缩放，然后在两个“相反”方向中的一个方向上离开，最终形成一个奇点或分散。弛豫和偏离吸引子是由准正模和一个生长模在线性近似中控制的。此外，在数值演化（非线性偏微分方程）中，已经观察到用特征值问题（线性偏微分方程）计算的扰动模态及其振荡频率和阻尼或放大因子的分布。特别是，生长模式的指数决定了中间动力学的寿命（在吸引子附近），并且在松弛过程中观察到一些最小阻尼的qnm为振铃。
We study the dynamics of nonlinear wave equations near the threshold of singularity formation, motivated by some unexpected features observed in the critical gravitational collapse. One of the features, the existence of a universal intermediate attractor in the dynamics at the threshold of singularity formation, is studied in more detail. The result is a scheme of the evolution: for all initial data sufficiently near the threshold, the solutions reach one universal static solution, which plays the role of an intermediate attractor. The solutions remain near the attractor for some time, which scales in a simple way with a parameter of the initial data, and then departs in one of two “opposite” directions, eventually forming a singularity or dispersing. The relaxation as well as the departure from the attractor is governed in the linear approximation by quasi-normal modes (QNMs) and one growing mode. Moreover, the profiles of the perturbation modes with their oscillation frequencies and damping or amplifying factors, calculated as eigenvalue problems (linear ODEs) have been observed in numerical evolution (nonlinear PDEs). In particular, the exponent of the growing mode determines the lifetime of the intermediate dynamics (near the attractor), and a few least damped QNMs have been observed as ringing in the relaxation process.