arXiv math/97122781997-12-12math-phmath.MP

Ginzburg-Landau-薛定谔方程的涡旋动力学

Vortex Dynamics for the Ginzburg-Landau-Schrödinger Equation

James Ellis Colliander, Robert L. Jerrard

摘要：

在对初始数据$ϕ^ε$的两个主要假设下，研究了Ginzburg-Landau-薛定谔方程在$ε\rightarrow 0$极限下的初值问题.第一个假设是$ϕ^ε$呈现出$m$不同的$\pm 1$度的旋涡;这些旋涡被描述为$ϕ^ε$的雅可比矩阵$[Jϕ^ε]$的集中点。其次，我们假设能量边界与集中点处的旋涡一致。在这些假设下，我们确定“涡旋结构”的$ε\rightarrow 0$极限的$ϕ^ε$和这些结构持续的解决方案$u^ε(t)$的$GLS_ε$。我们推导出常微分方程，支配运动的旋涡在$ε\rightarrow 0$限制。常微分方程的极限系统是由Bethuel、Brezis和Hélein的重整化能量支配的哈密顿流。我们的论点依赖于在另一篇论文中证明的涡的结构稳定性的结果。

英文摘要：

The initial value problem for the Ginzburg-Landau-Schrödinger equation is examined in the $ε\rightarrow 0$ limit under two main assumptions on the initial data $ϕ^ε$. The first assumption is that $ϕ^ε$ exhibits $m$ distinct vortices of degree $\pm 1$; these are described as points of concentration of the Jacobian $[Jϕ^ε]$ of $ϕ^ε$. Second, we assume energy bounds consistent with vortices at the points of concentration. Under these assumptions, we identify ``vortex structures'' in the $ε\rightarrow 0$ limit of $ϕ^ε$ and show that these structures persist in the solution $u^ε(t)$ of $GLS_ε$. We derive ordinary differential equations which govern the motion of the vortices in the $ε\rightarrow 0$ limit. The limiting system of ordinary differential equations is a Hamitonian flow governed by the renormalized energy of Bethuel, Brezis and Hélein. Our arguments rely on results about the structural stability of vortices which are proved in a separate paper.

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摘要：

英文摘要：

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