To Snake or Not to Snake in the Planar Swift-Hohenberg Equation
To Snake or Not to Snake in the Planar Swift-Hohenberg Equation
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DOI：
10.1137/100782747
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发表时间：
2010-01-01
影响因子：
2.1
通讯作者：
Sandstede, Bjoern
中科院分区：
文献类型：
--
作者：
Avitabile, Daniele;Lloyd, David J. B.;Sandstede, Bjoern
研究了二维Swift-Hohenberg方程在无限长圆柱体和平面上的定域图的分岔结构。在圆柱体上，我们发现局部的滚动、正方形和条纹斑块在相同的分岔分支上表现出蛇形和非蛇形行为。其中一些图案在四个鞍结界限之间蜿蜒；在这种情况下，最近的分析结果预测了不对称解存在丰富的分岔结构，并追踪了这些分支和沿着这些分支的PDE谱。在平面上，我们研究了通常被称为蠕虫的完全定域滚动结构的分岔结构。在上述所有情况下，我们使用几何思想和空间动力学技术来解释我们遇到的现象。
We investigate the bifurcation structure of stationary localized patterns of the two-dimensional Swift-Hohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localized roll, square, and stripe patches that exhibit snaking and nonsnaking behavior on the same bifurcation branch. Some of these patterns snake between four saddle-node limits; in this case, recent analytical results predict the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localized roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena that we encounter.