## Dynamics of Ginzburg-Landau vortices.(English)Zbl 0923.35167

The authors study the asymptotics of the sequence of complex valued solutions $$u^\varepsilon(x,t)$$ in the limit $$\varepsilon\to 0$$ of the system $\partial u^\varepsilon/\partial t-\Delta u^\varepsilon= \varepsilon^{-2} u^\varepsilon(1-| u^\varepsilon|^2)\quad\text{in }\Omega\times (0,\infty),$
$u^\varepsilon(x,t)= g(x)\quad\text{on }\partial\Omega\times (0,\infty).$ Here $$\Omega\subset \mathbb{R}^2$$ is an open bounded subset and $$g$$ is a given function with $$| g|= 1$$. The system provides a gradient flow of the functional $I^\varepsilon(w)= \int_\Omega (\textstyle{{1\over 2}}|\nabla w|^2+ \varepsilon^{-2} \textstyle{{1\over 4}}(1-| w|^2))dx$ which serves for the main technical tool. The most interesting geometrical results concern the behaviour of vortices (the zeroes of solutions). Assuming that initially there are $$N$$ isolated vortices with degree $$\pm 1$$, then, in the limit, these vortices persist and satisfy a system of ordinary differential equations of the kind ${d\over dt} y^i(t)= -2d_i\Biggl((\nabla\varphi(y^i(t), \vec y(t)))^\perp+ \sum_{m\neq i} d_m {y^m(t)- y^i(t)\over| y^m(t)- y^i(t)|^2}\Biggr),$ where $$d_i\in\{\pm 1\}$$, $$\varphi$$ is a solution of a Dirichlet problem (which cannot be stated here), $$\perp$$ denotes an orthogonal vector.
All the proofs are given with details and are completed by numerous comments.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35K57 Reaction-diffusion equations
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