Diffusive mixing of periodic wave trains in reaction–diffusion systems
Diffusive mixing of periodic wave trains in reaction–diffusion systems
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反应扩散系统中周期波列的扩散混合
DOI：
10.1016/j.jde.2011.10.014
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发表时间：
2011
影响因子：
2.4
通讯作者：
H. Uecker
中科院分区：
文献类型：
--
作者：
Bjorn Sandstede;A. Scheel;G. Schneider;H. Uecker
我们考虑无限线上的反应扩散系统，其表现出一系列由波数k参数化的光谱稳定的空间周期波列u0（kx−ωt;k）。我们证明了对于初始收敛于这些状态为x→±∞的解，在无穷远处具有不同相位φ -≠φ +的渐近状态u0（kx+ φ±；k）为x→±∞的稳定扩散混合。证明是基于布洛赫波分析，重整化理论，并严格分解这些波解的扰动到一个相位模式，它显示扩散行为，和一个指数阻尼剩余。根据色散关系的不同，渐近态可以与最低阶高斯分布线性混合，也可以与非对称非高斯分布线性混合，该分布由Burgers方程给出，即非平凡色散关系下扩散模的振幅方程。
We consider reaction–diffusion systems on the infinite line that exhibit a family of spectrally stable spatially periodic wave trains u0(kx−ωt;k) that are parameterized by the wave number k. We prove stable diffusive mixing of the asymptotic states u0(kx+ϕ±;k) as x→±∞ with different phases ϕ−≠ϕ+at infinity for solutions that initially converge to these states as x→±∞. The proof is based on Bloch wave analysis, renormalization theory, and a rigorous decomposition of the perturbations of these wave solutions into a phase mode, which shows diffusive behavior, and an exponentially damped remainder. Depending on the dispersion relation, the asymptotic states mix linearly with a Gaussian profile at lowest order or with a nonsymmetric non-Gaussian profile given by Burgers equation, which is the amplitude equation of the diffusive modes in the case of a nontrivial dispersion relation.