Nonlinear stability of source defects in the complex Ginzburg–Landau equation
Nonlinear stability of source defects in the complex Ginzburg–Landau equation
复制标题
复杂 Ginzburg-Landau 方程中源缺陷的非线性稳定性
DOI：
10.1088/0951-7715/27/4/739
复制
发表时间：
2013
期刊：
影响因子：
1.7
通讯作者：
K. Zumbrun
中科院分区：
文献类型：
--
作者：
M. Beck;Toan T. Nguyen;Bjorn Sandstede;K. Zumbrun
在适当的移动坐标系中，源缺陷是反应扩散方程的时间周期解，其在空间上渐近于群速度指向远离缺陷核心的空间周期波列。在本文中，我们严格地建立非线性稳定性的光谱稳定源缺陷的复Ginzburg-Landau方程。由于远场的向外传输，局部扰动可能导致高度非局部化的响应，即使在线性水平上。为了克服这一点，我们首先详细研究了线性化方程的解的动力学。这使我们能够确定一个近似的解决方案，满足完整的方程，并包括二次项的非线性。这种近似利用了这样一个事实，即非本地化的相位响应，从嵌入的零特征值，可以捕获，领先的顺序，由非线性Burgers方程。分析是通过获得详细的估计的预解核和逐点估计的绿色的功能，它允许一个关闭的非线性迭代计划。
In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction–diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg–Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for Green's function, which allow one to close a nonlinear iteration scheme.