论文标题
复杂的金茨堡 - 兰道方程中图案形成前线的光谱稳定性，具有淬火机制
Spectral stability of pattern-forming fronts in the complex Ginzburg-Landau equation with a quenching mechanism
论文作者
论文摘要
我们考虑了复杂的金茨堡 - 兰道方程中的图案形成前线，其空间异质性在域中逐渐稳定或淬灭时，在域中稳定或淬灭了微不足道的基态。我们考虑的是，异质性以速度$ c $低于相关均匀系统中图案形成前线的线性入侵速度的速度传播。在这种情况下，前部锁定了异质性的界面，使长期的中间状态位于不稳定的基态状态附近，可能允许扰动的生长。通过特征值的积累到与不稳定的基态相关的绝对频谱上，这表现在正面的线性化光谱中。随着淬火速度$ c $的线性入侵速度增加，绝对频谱稳定在其上积累的速率相同的速度，使我们能够在$ l^2（\ mathbb {r}）中严格地建立前沿的频谱稳定性。 不稳定的绝对光谱的存在构成了技术挑战，因为中间状态的空间特征值不再承认双曲线分裂和标准工具（例如指数二分法）不可用。取而代之的是，我们将线性流程进行投影，并将Riemann表面与叠加原理结合起来，以研究子空间的演变，作为Grassmannian歧管上相关矩阵Riccati微分方程的解决方案。然后，特征值可以被识别为Meromorormormormorphic riccati-evans功能的根，并且可以使用绕组数量和奇偶校验参数位于位置。
We consider pattern-forming fronts in the complex Ginzburg-Landau equation with a traveling spatial heterogeneity which destabilizes, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed $c$ just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearization about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed $c$ increases towards the linear invasion speed, the absolute spectrum stabilizes with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectrally stability of the front in $L^2(\mathbb{R})$. The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivize the linear flow, and use Riemann surface unfolding in combination with a superposition principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati-Evans function, and can be located using winding number and parity arguments.
