Vortex Dynamics for the Ginzburg-Landau-Schrödinger Equation

Abstract

The initial value problem for the Ginzburg-Landau-Schrödinger equation is examined in the $\epsilon \to 0$ limit under two main assumptions on the initial data ${\varphi }^{\epsilon }$. The first assumption is that ${\varphi }^{\epsilon }$ exhibits $m$ distinct vortices of degree $\pm 1$; these are described as points of concentration of the Jacobian $[J{\varphi }^{\epsilon }]$ of ${\varphi }^{\epsilon }$. Second, we assume energy bounds consistent with vortices at the points of concentration. Under these assumptions, we identify ``vortex structures'' in the $\epsilon \to 0$ limit of ${\varphi }^{\epsilon }$ and show that these structures persist in the solution $u^{\epsilon }(t)$ of $GL{S}_{\epsilon }$. We derive ordinary differential equations which govern the motion of the vortices in the $\epsilon \to 0$ 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.

Cited by in corpus (7)

• Unstable manifolds and Schroedinger dynamics of Ginzburg-Landau vortices
• Scattering theory for the Gross-Pitaevskii equation in three dimensions
• Vortices in a Stochastic Parabolic Ginzburg-Landau Equation
• Dynamics of nearly parallel vortex filaments for the Gross-Pitaevskii equation
• Hydrodynamic limit of the Gross-Pitaevskii equation
• Numerical Study of Quantized Vortex Interaction in Complex Ginzburg--Landau Equation on Bounded Domains
• On the linear wave regime of the Gross-Pitaevskii equation