Abstract
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.
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References
M. Levin and X.-G. Wen, Rev. Modern Phys., 77, 871–879 (2005); arXiv:cond-mat/0407140v2 (2004).
W. Thomson, Phil. Mag., 34, 15–24 (1867).
K. Moffatt, Rus. J. Nonlin. Dynamics, 2, 401–410 (2006).
F. Wilczek, Phys. Rev. Lett., 48, 1144–1146 (1982).
G. Moore, Nucl. Phys. B., 360, 362–396 (1991).
A. P. Protogenov, Sov. Phys. Usp., 35, 535–571 (1992).
M. G. Alford and F. Wilczek, Phys. Rev. Lett., 62, 1071–1074 (1989).
S. V. Talalov, Theor. Math. Phys., 165, 1517–1526 (2010).
S. V. Talalov, Theor. Math. Phys., 152, 1234–1242 (2007).
P. P. Kulish and A. G. Reiman, J. Soviet Math., 22, 1627–1637 (1983).
L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Methods in the Theory of Solitons [in Russian], Nauka, Moscow (1986); English transl., Springer, Berlin (1987).
P. G. Saffman, Vortex Dynamics, Cambridge Univ. Press., Cambridge (1992).
S. V. Alekseenko, P. A. Kuibin, and V. L. Okulov, Introduction to the Theory of Concentrated Vortices [in Russian], Inst. Heat Physics, Siberian Branch, Russ. Acad. Sci., Novosibirsk (2003).
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Univ. Press, Cambridge (1967).
N. Ya. Vilenkin, Special Functions and the Theory of Representations of Groups [in Russian], Nauka, Moscow (1965); English transl. (Transl. Math. Monogr., Vol. 22), Amer. Math. Soc., Providence, R. I. (1968).
V. I. Fushchich and A. G. Nikitin, Symmetries of Equations of Quantum Mechanics [in Russian], Nauka, Moscow (1990); English transl., Allerton Press, New York (1994).
R. Jackiw and V. P. Nair, Phys. Lett. B, 480, 237–238 (2000); arXiv:hep-th/0003130v2 (2000).
J. Negro, M. A. del Olmo, and J. Tosiek, J. Math. Phys., 47, 033508 (2006); arXiv:math-ph/0512007v3 (2005).
S. V. Talalov, Internat. J. Mod. Phys. A., 26, 2757–2772 (2011); arXiv:1105.0743v1 [math-ph] (2011).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 176, No. 3, pp. 372–384, September 2013.
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Talalov, S.V. Solutions of string, vortex, and anyon types for the hierarchy of the nonlinear Schrödinger equation. Theor Math Phys 176, 1145–1155 (2013). https://doi.org/10.1007/s11232-013-0095-0
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DOI: https://doi.org/10.1007/s11232-013-0095-0