Invariant 2-tori in the time-dependent Ginzburg-Landau equation
Invariant 2-tori in the time-dependent Ginzburg-Landau equation
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DOI：
10.1088/0951-7715/5/2/002
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发表时间：
1992-03
期刊：
影响因子：
1.7
通讯作者：
P. Takáč
中科院分区：
文献类型：
--
作者：
P. Takáč
数值结果表明，在一维周期边界条件下，含时复Ginzburg-Landau方程（CGL）存在2-和3-环面。作者采用标准的分歧方法，如Lyapunov-Schmidt约化，通过严格的分析工具证明了2-环面的存在性。分叉涉及多维（真实的或复）特征空间。该方法可以得到一个唯一的光滑分支流形，其维数等于特征空间的维数。特别是，从零解的主要分叉产生周期轨道，从旋转波的二次分叉产生2-环面。利用中心流形理论讨论了它们的稳定性。虽然旋转波由具有时间和空间恒定模量的波函数表示，但分叉2-环面具有波函数，其模量是行波。在超导性中，波函数的平方模与超导电子的密度成正比。
Numerical evidence shows the existence of 2- and 3-tori in the time-dependent complex Ginzburg-Landau equation (CGL) in one spatial dimension with periodic boundary conditions. The author employs standard bifurcation methods, such as the Lyapunov-Schmidt reduction, to prove the existence of 2-tori by rigorous analytical tools. Bifurcations involve multi-dimensional (real or complex) eigenspaces. The procedure enables one to obtain a unique smooth bifurcating manifold the dimension of which equals the dimension of the eigenspace. In particular, primary bifurcations from the zero solution yield periodic orbits, and secondary ones from rotating waves yield 2-tori. Their stability is discussed by means of centre manifold theory. While the rotating waves are represented by a wavefunction with temporally and spatially constant modulus, the bifurcating 2-tori have a wavefunction the modulus of which is a travelling wave. In superconductivity the square modulus of the wavefunction is proportional to the density of superconducting electrons.