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.


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