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Bethuel, F.; Orlandi, G.; Smets, D.

Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics. (English) Zbl 1087.35008

Duke Math. J. 130, No. 3, 523-614 (2005).
The paper aims to establish a rigorous mathematical setting for the usual two-dimensional Ginzburg-Landau equation with real coefficients, in the form of \[ \varepsilon\left(u_t - \nabla^2u\right) = u\left(1 - | u| ^2\right), \] where \(\varepsilon\) is treated as a formal small parameter. An evolution problem is considered, initiated by a configuration containing several fundamental vortices with topological charges \(\pm 1\) (evolution of unstable vortices with a multiple topological charge is not considered in the paper) and a plane wave. The evolution includes collisions between vortices, and drift of individual vortices driven by the plane wave. The proofs of theorems about the existence of the evolution solutions are based on separation between dynamical effects taking place at different time scales. The existence of solutions is established both before and after collisions between the vortices. In particular, it is proven that the evolution of a configuration including equal numbers of the fundamental vortices and antivortices leads, in the generic case, to complete annihilation of the vortices.
Reviewer: Boris A. Malomed (Tel Aviv)

Cited in 18 Documents

MSC:

35B25 Singular perturbations in context of PDEs
35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

vortex annihilation; vortex drift; multiple-scale expansion; two space dimensions; different time scales
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References:

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