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 TMF, 1998, Volume 115, Number 3, Pages 323–348 (Mi tmf877)

Nonlinear $\sigma$-model in a curved space, gauge equivalence, and exact solutions of $(2+0)$-dimensional integrable equations

E. Sh. Gutshabasha, V. D. Lipovskii, S. S. Nikulichev

a V. A. Fock Institute of Physics, Saint-Petersburg State University

Abstract: We propose a nonlinear $\sigma$-model in a curved space as a general integrable elliptic model. We construct its exact solutions and obtain energy estimates near the critical point. We consider the Pohlmeyer transformation in Euclidean space and investigate the gauge equivalence conditions for a broad class of elliptic equations. We develop the inverse scattering transform method for the $\operatorname {sh}$-Gordon equation and evaluate its exact and asymptotic solutions.

DOI: https://doi.org/10.4213/tmf877

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English version:
Theoretical and Mathematical Physics, 1998, 115:3, 619–638

Bibliographic databases:

Citation: E. Sh. Gutshabash, V. D. Lipovskii, S. S. Nikulichev, “Nonlinear $\sigma$-model in a curved space, gauge equivalence, and exact solutions of $(2+0)$-dimensional integrable equations”, TMF, 115:3 (1998), 323–348; Theoret. and Math. Phys., 115:3 (1998), 619–638

Citation in format AMSBIB
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\by E.~Sh.~Gutshabash, V.~D.~Lipovskii, S.~S.~Nikulichev
\paper Nonlinear $\sigma$-model in a curved space, gauge equivalence, and exact solutions of
$(2+0)$-dimensional integrable equations
\jour TMF
\yr 1998
\vol 115
\issue 3
\pages 323--348
\mathnet{http://mi.mathnet.ru/tmf877}
\crossref{https://doi.org/10.4213/tmf877}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1692414}
\zmath{https://zbmath.org/?q=an:1113.81095}
\elib{http://elibrary.ru/item.asp?id=13278930}
\transl
\jour Theoret. and Math. Phys.
\yr 1998
\vol 115
\issue 3
\pages 619--638
\crossref{https://doi.org/10.1007/BF02575486}

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• http://mi.mathnet.ru/eng/tmf/v115/i3/p323

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This publication is cited in the following articles:
1. Pritula, GM, “Stationary structures in two-dimensional continuous Heisenberg ferromagnetic spin system”, Journal of Nonlinear Mathematical Physics, 10:3 (2003), 256
2. E. Sh. Gutshabash, “Hydrodynamical vortice on the plain”, J. Math. Sci. (N. Y.), 143:1 (2007), 2765–2772
3. Mehrabi, AR, “ANALYSIS AND SIMULATION OF LONG-RANGE CORRELATIONS IN CURVED SPACE”, International Journal of Modern Physics C, 20:8 (2009), 1211
4. E. Sh. Gutshabash, “Nonlinear sigma model, Zakharov–Shabat method, and new exact forms of the minimal surfaces in ${\mathbb R}^3$”, JETP Letters, 99:12 (2014), 715–719
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