On the Ginzburg-Landau and related equations

Yu N. Ovchinnikov[1]; Israel Michael Sigal[2]

  • [1] L.D. Laudau Institute, Moscow
  • [2] Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3 Canada

Séminaire Équations aux dérivées partielles (1997-1998)

  • Volume: 1997-1998, page 1-13

Abstract

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We describe qualitative behaviour of solutions of the Gross-Pitaevskii equation in 2D in terms of motion of vortices and radiation. To this end we introduce the notion of the intervortex energy. We develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory. Finally we mention briefly two special situations where we are able to obtain rather detailed picture of the vortex dynamics. Our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized, stable static solutions. It yields description of general time-dependent solutions in terms of dynamics of those static solutions “glued” together.

How to cite

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Ovchinnikov, Yu N., and Sigal, Israel Michael. "On the Ginzburg-Landau and related equations." Séminaire Équations aux dérivées partielles 1997-1998 (1997-1998): 1-13. <http://eudml.org/doc/10948>.

@article{Ovchinnikov1997-1998,
abstract = {We describe qualitative behaviour of solutions of the Gross-Pitaevskii equation in 2D in terms of motion of vortices and radiation. To this end we introduce the notion of the intervortex energy. We develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory. Finally we mention briefly two special situations where we are able to obtain rather detailed picture of the vortex dynamics. Our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized, stable static solutions. It yields description of general time-dependent solutions in terms of dynamics of those static solutions “glued” together.},
affiliation = {L.D. Laudau Institute, Moscow; Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3 Canada},
author = {Ovchinnikov, Yu N., Sigal, Israel Michael},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {time-dependent Ginzburg-Landau equation; Gross-Pitaevskii equation; intervortex energy; vortex dynamics; stable static solutions},
language = {eng},
pages = {1-13},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {On the Ginzburg-Landau and related equations},
url = {http://eudml.org/doc/10948},
volume = {1997-1998},
year = {1997-1998},
}

TY - JOUR
AU - Ovchinnikov, Yu N.
AU - Sigal, Israel Michael
TI - On the Ginzburg-Landau and related equations
JO - Séminaire Équations aux dérivées partielles
PY - 1997-1998
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 1997-1998
SP - 1
EP - 13
AB - We describe qualitative behaviour of solutions of the Gross-Pitaevskii equation in 2D in terms of motion of vortices and radiation. To this end we introduce the notion of the intervortex energy. We develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory. Finally we mention briefly two special situations where we are able to obtain rather detailed picture of the vortex dynamics. Our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized, stable static solutions. It yields description of general time-dependent solutions in terms of dynamics of those static solutions “glued” together.
LA - eng
KW - time-dependent Ginzburg-Landau equation; Gross-Pitaevskii equation; intervortex energy; vortex dynamics; stable static solutions
UR - http://eudml.org/doc/10948
ER -

References

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