Geodesic
Solutions of string, vortex, and anyon types for the hierarchy of the nonlinear Schrödinger equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 176 (2013) no. 3, pp. 372-384 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the relation that associates an infinite curve evolving in the $E_3$ space with each equation of the hierarchy of the nonlinear Schrödinger and the modified Korteveg–de Vries equations. We show that the low levels of the hierarchy correspond to known objects, which are strings and vortex lines endowed with various structures. We consider one of the hierarchy levels corresponding to the dynamics of the vortex line in the local induction approximation in detail. We construct the Hamiltonian description of the corresponding dynamics admitting an interpretation in terms of a quasiparticle in a plane, the "anyon." We propose a scheme for quantizing the theory in a framework in which we obtain a formula for a (generally fractional) spin value.
Keywords: hierarchy of the nonlinear Schrödinger equation, modified Korteveg–de Vries equation, anyon.
Mots-clés : vortex 
@article{TMF_2013_176_3_a2,
     author = {S. V. Talalov},
     title = {Solutions of string, vortex, and anyon types for the~hierarchy of the~nonlinear {Schr\"odinger} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {372--384},
     year = {2013},
     volume = {176},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_176_3_a2/}
}
TY  - JOUR
AU  - S. V. Talalov
TI  - Solutions of string, vortex, and anyon types for the hierarchy of the nonlinear Schrödinger equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2013
SP  - 372
EP  - 384
VL  - 176
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2013_176_3_a2/
LA  - ru
ID  - TMF_2013_176_3_a2
ER  - 
%0 Journal Article
%A S. V. Talalov
%T Solutions of string, vortex, and anyon types for the hierarchy of the nonlinear Schrödinger equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2013
%P 372-384
%V 176
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2013_176_3_a2/
%G ru
%F TMF_2013_176_3_a2
S. V. Talalov. Solutions of string, vortex, and anyon types for the hierarchy of the nonlinear Schrödinger equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 176 (2013) no. 3, pp. 372-384. http://geodesic.mathdoc.fr/item/TMF_2013_176_3_a2/

[1] M. Levin, X.-G. Wen, Rev. Modern Phys., 77:3 (2005), 871–879, arXiv: cond-mat/0407140 | DOI

[2] W. Thomson, Phil. Mag., 34 (1867), 15–24

[3] K. Moffatt, Nelineinaya dinam., 2:4 (2006), 401–410

[4] F. Wilczek, Phys. Rev. Lett., 48:17 (1982), 1144–1146 | DOI

[5] G. Moore, Nucl. Phys. B., 360:2–3 (1991), 362–396 | DOI | MR

[6] A. P. Protogenov, UFN, 162:7 (1992), 1–80 | DOI | DOI

[7] M. G. Alford, F. Wilczek, Phys. Rev. Lett., 62:10 (1989), 1071–1074 | DOI | MR

[8] S. V. Talalov, TMF, 165:2 (2010), 329–340 | DOI | DOI | Zbl

[9] S. V. Talalov, TMF, 152:3 (2007), 430–439 | DOI | DOI | MR | Zbl

[10] P. P. Kulish, A. G. Reiman, Zap. nauchn. sem. LOMI, 77 (1978), 134–147 | MR | Zbl

[11] L. A. Takhtadzhyan, L. D. Faddeev, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | MR | Zbl | Zbl

[12] F. Dzh. Seffmen, Dinamika vikhrei, Nauchnyi mir, M., 2000 | MR | Zbl

[13] S. V. Alekseenko, P. A. Kuibin, V. L. Okulov, Vvedenie v teoriyu kontsentrirovannykh vikhrei, In-t teplofiziki SO RAN, Novosibirsk, 2003 | MR | Zbl

[14] Dzh. Betchelor, Vvedenie v dinamiku zhidkosti, Mir, M., 1973 | MR | Zbl

[15] N. Ya. Vilenkin, Spetsialnye funktsii i teoriya predstavlenii grupp, Nauka, M., 1965 | MR | Zbl

[16] V. I. Fuschich, A. G. Nikitin, Simmetriya uravnenii kvantovoi mekhaniki, Nauka, M., 1990 | MR | MR | Zbl | Zbl

[17] R. Jackiw, V. P. Nair, Phys. Lett. B, 480:1–2 (2000), 237–238, arXiv: hep-th/0003130 | DOI | MR | Zbl

[18] J. Negro, M. A. del Olmo, J. Tosiek, J. Math. Phys., 47:3 (2006), 033508, 19 pp., arXiv: math-ph/0512007 | DOI | MR | Zbl

[19] S. V. Talalov, Internat. J. Modern Phys. A., 26:16 (2011), 2757–2772, arXiv: 1105.0743 | DOI | MR | Zbl