On the Ginzburg-Landau and related equations
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1997-1998), Talk no. 21, 13 p.

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.

Ovchinnikov, Yu N. 1; Sigal, Israel Michael 2

1 L.D. Laudau Institute, Moscow
2 Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3 Canada
     author = {Ovchinnikov, Yu N. and Sigal, Israel Michael},
     title = {On the {Ginzburg-Landau} and related equations},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:21},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1997-1998},
     mrnumber = {1660534},
     zbl = {1061.35522},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_1997-1998____A21_0/}
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AU  - Sigal, Israel Michael
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JO  - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz"
N1  - talk:21
PY  - 1997-1998
PB  - Centre de mathématiques Laurent Schwartz, École polytechnique
UR  - http://www.numdam.org/item/SEDP_1997-1998____A21_0/
LA  - en
ID  - SEDP_1997-1998____A21_0
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%D 1997-1998
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%U http://www.numdam.org/item/SEDP_1997-1998____A21_0/
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Ovchinnikov, Yu N.; Sigal, Israel Michael. On the Ginzburg-Landau and related equations. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (1997-1998), Talk no. 21, 13 p. http://www.numdam.org/item/SEDP_1997-1998____A21_0/

[1] F. Bethuel, H. Brezis and F. Hélein (1994), Ginzburg-Landau Vortices, Birkhäuser, Basel. | MR | Zbl

[2] F. Bethuel and J.C. Saut (1997), Travelling waves for the Gross-Pitrevskii equation, preprint.

[3] S. Chanillo and M. Kiesling (1995), Symmetry of solutions of Ginzburg-Landau equations, Compt. Rend. Acad. Sci. Paris, t. 327, Série I, 1023–1026. | MR | Zbl

[4] Y. Chen, C. Elliot and T. Qui (1994), Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation, Proc. Royal Soc. Edinburgh 124A, 1068-1075. | MR | Zbl

[5] J.E. Colliander and R.L. Jerrard (1998), Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, preprint, MSRI. | MR

[6] J. Creswick and N. Morrison (1980), On the dynamics of quantum vortices, Phys. Lett. A 76, 267. | MR

[7] W. E (1994), Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Physica D 77, 383-404. | MR | Zbl

[8] P. Fife and L.A. Peletier (1996), On the location of defects in stationary solutions of the Ginzburg-Landau equations on R 2 , Quart. Appl. Math. 54, 85-104. | MR | Zbl

[9] J. Fröhlich and M. Struwe (1990), Variational problems on vector bundles, Commun. Math. Phys. 131, 431-464. | MR | Zbl

[10] V.L. Ginzburg and L.P. Pitaevskii (1958), On the theory of superfluidity, Sov. Phys. JETP 7, 585. | MR

[11] E.P. Gross (1961), Nuovo Cimento A 20, 454. | MR | Zbl

[12] E. Gross (1966), Dynamics of interacting bosons, in Physics of Many Particle Systems, ed. E. Meeron, Gordon and Breach, NY, 268.

[13] S. Gustafson (1997a), Stability of vortex solutions of the Ginzburg-Landau heat equation, in PDE’s and their applications (L. Seco et al, eds), Proceeding of Conference in PDE’s, Toronto June 1995. | Zbl

[14] S. Gustafson (1997b), Symmetric solutions of hte Ginzburg-Landau in all dimensions, IMRN, 16, 807-816. | MR | Zbl

[15] S. Gustafson and I.M. Sigal (1998), Existence and stability of magnetic vortices. preprint (Toronto).

[16] P. Hagan (1982), Spiral waves in reaction diffusion equations, SIAM J. Applied Math. 42, 762–786. | MR | Zbl

[17] M. Hervé, R. Hervé (1994), Étude qualitative des solutions réeles d’une équation différentielle liée a l’équation de Ginzburg-Landau, Ann. Inst. Henri Poincaré, Analyse non linéaire 11, 427-440. | EuDML | Numdam | MR | Zbl

[18] S.V. Iordanskii and A.V. Smirnov (1978), JETP Lett. 27, 535.

[19] A. Jaffe and C. Taubes (1980), Vortices and Monopoles, Birkhäuser. | MR | Zbl

[20] C. Jones, S.J. Putterman and P.M. Roberts (1986), Motion of Bose condensation V, J. Phys. A 19, 2991–3011.

[21] C.A. Jones and P.M. Roberts (1982), J. Phys. A: Math. Gen. 15, 2599–2619.

[22] E.A. Kuznetzov and J.J. Rasmussen (1995), Instability of two dimensional solitons and vortices in defocusing media, Phys. Rev. E 51, 5, 4479–4484.

[23] E.M. Lieb and M. Loss (1994), Symmetry of the Ginzburg-Landau minimizers in a disc, Math. Res. Lett. 1, 701–715. | MR | Zbl

[24] F.-H. Lin and J.X. Xin (1998), On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation, preprint. | MR | Zbl

[25] N.S. Manton (1981), A remark on scattering of BPS monopoles, Phys. Letters 110B, N1, 54–56. | MR

[26] P. Mironescu (1995), On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal. 130, 334–344. | MR | Zbl

[27] P. Mironescu (1996), Les minimiseurs locaux pour l’équation de Ginzburg-Landau sont à symmétrie radiale, preprint. | MR

[28] J. Neu (1990), Vortices in complex scalar fields, Physica D 43, 385–406. | MR | Zbl

[29] L. Onsager (1949), Statistical hydrodynamics, Nuovo Cimento V-VI, Suppl. 2, 279. | MR

[30] Yu.N. Ovchinnikov and I.M. Sigal (1997a), Ginzburg-Landau equation I. General discussion, in P.D.E.’s and their Applications (L. Seco et al., eds.), Proceedings of Conference in PDE’s, Toronto, June 1995. | Zbl

[31] Yu.N. Ovchinnikov and I.M. Sigal (1997b), The Ginzburg-Landau equation II. The energy of vortex configurations, preprint.

[32] Yu.N. Ovchinnikov and I.M. Sigal (1998a), The Ginzburg-Landau equation III. Vortex dynamics, Nonlinearity (to appear). | MR | Zbl

[33] Yu.N. Ovchinnikov and I.M. Sigal (1998b), Symmetry breaking in the Ginzburg-Landau equation, preprint.

[34] Yu.N. Ovchinnikov and I.M. Sigal (1998c), Long-time behaviour of Ginzburg-Landau vortices, Nonlinearity (to appear). | MR | Zbl

[35] Yu.N. Ovchinnikov and I.M. Sigal (1998d), Break up and creation of vorticies, in preparation.

[36] L.M. Pismen (1994), Structure and dynamics of defects in 2D complex vector field, Physica D 73, 244-258. | MR | Zbl

[37] L.M. Pismen and A. Nepomnyashchy (1993), Stability of vortex rings in a model of superflow, Physica D 69, 163–171. | MR | Zbl

[38] L.P. Pitaevskii (1961), Pis’ma Zh. Eksp. Teor. Fix. 77, 988 (Sov. Phys. JETP 13, 451).

[39] A.S. Schwarz (1993), Topology for physicists, Springer-Verlag. | MR | Zbl

[40] I. Shafrir (1994), Remarks on solutions of -Δu=(1-|u| 2 )u in 2 , C.R. Acad. Sci. Paris, t. 318, Série I, 327–331. | MR | Zbl

[41] D. Stuart (1994), Comm. Math. Phys. 159, 51. | MR | Zbl

[42] G.B. Whitham (1974), Linear and Nonlinear Waves, John Wiley & Sons. | MR | Zbl