Steady-state bifurcation with Euclidean symmetry
Steady-state bifurcation with Euclidean symmetry
复制标题
具有欧氏对称性的稳态分岔
DOI：
10.1090/s0002-9947-99-02147-9
复制
发表时间：
1999
影响因子：
1.3
通讯作者：
I. Melbourne
中科院分区：
文献类型：
--
作者：
I. Melbourne
我们考虑在欧氏群E（n）下等变的偏微分方程系统从一个完全对称平衡点经历稳态分支（临界波数为非零）。一个严格的减少程序，导致当地的最佳小系统的方程。特别地，当n = 1和n = 2时，对于具有一般n的反应扩散方程，约化导致单个方程。(Our结果一般是有效的，扰动由相对有界的偏微分算子组成。类似于紧致群的等变分歧理论，我们根据E（n）的绝对不可约酉表示对不同类型的约化系统进行了分类. E（n）的表示理论是由O（n - 1）的不可约表示驱动的。当n = 1时，这就构成了金-朗道方程在直线上的“普适性”的数学陈述。(In最近的工作，我们解决了这个方程的有效性使用相关技术。当n = 2时，精确地有两种显著不同类型的约化方程：标量和伪标量，对应于O（1）的平凡和非平凡一维表示。对于每一个n等于或大于3，都有无穷多种可能性。
We consider systems of partial differential equations equivariant under the Euclidean group E ( n ) and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when n = 1 and n = 2 and for reaction-diffusion equations with general n , reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of E ( n ). The representation theory of E ( n ) is driven by the irreducible representations of O ( n - 1). For n = 1, this constitutes a mathematical statement of the `universality' of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When n = 2, there are precisely two significantly different types of reduced equation: scalar and pseudoscalar , corresponding to the trivial and nontrivial one-dimensional representations of O (1). There are infinitely many possibilities for each n equal to or greater than 3.