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Motion of vortex lines in the Ginzburg-Landau model. (English) Zbl 0728.35090
Summary: Equations of motion for vortex lines in the Ginzburg-Landau theory are derived. We construct asymptotic approximations for the complex order parameter which are valid in the core region and in the far field. Using the method of matched asymptotic expansions we show that, to leading order, the line moves in the binormal direction with a curvature- dependent velocity. We also consider the contribution of remote parts of the line, interaction between several vortex lines and interaction with external fields.

35Q40 PDEs in connection with quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
81V99 Applications of quantum theory to specific physical systems
Full Text: DOI
[1] Batchelor, G.K., Introduction to fluid mechanics, (1967), Cambridge Univ. Press Cambridge · Zbl 0152.44402
[2] Bodenschatz, E.; Pesch, W.; Kramer, L., Structure and dynamics of dislocations in anisotropic pattern-forming systems, Physica D, 32, 135-145, (1988) · Zbl 0645.76090
[3] Glaberson, W.I.; Donnelly, R.J., Structure, distribution and dynamics of vortices in helium II, ()
[4] Hasimoto, H., A soliton on a vortex filament, J. fluid mech., 51, 477-485, (1972) · Zbl 0237.76010
[5] Kelvin, Lord, Mathematical and physical papers, Vol. IV, (1910), Cambridge · JFM 37.0766.01
[6] Kuramoto, Y., Chemical oscillations, waves and turbulence, (1984), Springer Berlin · Zbl 0558.76051
[7] Landau, L.D.; Lifshitz, E.M., Statistical physics, (1980), Pergamon Press London · Zbl 0080.19702
[8] Mermin, N.D., The topological theory of defects in ordered media, Rev. mod. phys., 51, 591-648, (1979)
[9] Neu, J., Vortices in complex scalar fields, Physica D, 43, 385-406, (1990) · Zbl 0711.35024
[10] Newton, P.; Keller, J.B., Stability of plane waves, SIAM J. appl. math., 47, 959-964, (1987) · Zbl 0644.35011
[11] L.M. Pismen and J.D. Rodriguez, Mobility of singularities in dissipative Ginzburg-Landau equation, Phys. Rev. A, in press.
[12] Pumir, A.; Siggia, E.D., Vortex dynamics and the existence of solutions to the Navier Stokes equations, Phys. fluids, 30, 1606-1626, (1987) · Zbl 0628.76033
[13] Rubinstein, J.; Sternberg, P.; Keller, J.B., Fast reaction, slow diffusion and curve shortening, SIAM J. appl. math, 49, 116-133, (1989) · Zbl 0701.35012
[14] J. Rubinstein, Self-induced motion of line defects, Quart. Appl. Math., in press. · Zbl 0728.35118
[15] Saffman, P.G.; Baker, G.R., Vortex interactions, Ann. rev. fluid mech., 11, 95-172, (1979) · Zbl 0434.76001
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