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V. D. Gershun

Publications and source records attributed to V. D. Gershun.

9 recordsLinked to original sources
arXiv2014

Integrable hydrodynamic equations for initial chiral currents and infinite hydrodynamic chains from WZNW model and string model of WZNW type with $SU(2),$ $SO(3),$ $SP(2),$ $SU(\infty)$, $SO(\infty)$, $SP(\infty)$ constant torsions

The WZNW and string models are considered in the terms of the initial and invariant chiral currents assuming that the internal and external torsions coincide (anticoincide) and they are the structure constants of the $SU(n),SO(n),$ $SP(n)$ Lie algebras. These models are the auxiliary problems in order to construct integrable equations of hydrodynamic type. It was shown that the WZNW and string models in terms of invariant chiral currents are integrable for the constant torsion associated with the structure constants of the $SU(2),$ $SO(3),$ $SP(2)$ and $SU(3)$ algebras only. The equation of motion for the density of the first Casimir operator was obtained in the form of the inviscid Burgers equation. The solution of this equation is presented through the Lambert function. Also, a new equation of motion for the initial chiral current was found. The integrable infinite hydrodynamic chains obtained from the WZNW and string models are given in terms of invariant chiral currents with the $SU(2)$, $SO(3)$, $SP(2)$ and with $SU(\infty)$, $SO(\infty)$, $SP(\infty)$ constant torsions. Also, the equations of motion for the density of any Casimir operator and new infinite dimensional equations of hydrodynamic type for the initial chiral currents through the symmetric structure constant of $SU(\infty)$, $SO(\infty),$ $SP(\infty)$ algebras are obtained.

nlin.SI↗
arXiv2005

Integrable string and hydrodynamical type models and nonlocal brackets

The closed string model in the background gravity field is considered as a bi-Hamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. The dual nonlocal Poisson brackets(PB), depending of the background fields and of their derivatives, are obtained. The integrability condition is formulated as the compatibility of the bi-Hamiltonity condition and the Jacobi identity of the dual PB. It is shown that the dual brackets and dual Hamiltonians can be obtained from the canonical PB and from the initial Hamiltonian by imposing the second kind constraints on the initial dynamical system, on the closed string model in the constant background fields, as example. The hydrodynamical type equation was obtained. Two types of the nonlocal brackets are introduced. Constant curvature and time-dependent metrics are considered. It is shown that the Jacobi identities for the nonlocal brackets have particular solution for the space-time coordinates, as matrix representation of the simple Lie group.

nlin.SI↗
arXiv2004

Nonlocal brackets and integrable string models

The closed string model in the background gravity field is considered as the bi-Hamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. The dual nonlocal Poisson brackets, de pending of the background fields and of there derivatives, are obtained. The integrability condition is formulated as compatibility of the bi-Hamiltonity con dition and the Jacobi identity of the dual Poisson bracket. Two types of the non local brackets are introduced. Constant curvature and time-dependent metrics are considered, as example. It is, shown that the Jacobi identities for the non local brackets have particular solution for the space-time coordinates, as matrix representation of the simple Lie group.

nlin.SI↗
arXiv2001

Bihamiltonian approach to the closed string model in the background fields

The closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system in assumption,that string model is the integrable model for particular kind of the background fields. It is shown, that bihamiltonity is origin of two types of the T-duality of the closed string models. The dual nonlocal Poisson brackets, depending of the background fields and of their derivatives, are obtained. The integrability condition is formulated as the compatibility of the bihamoltonity condition and the Jacobi identity of the dual Poisson bracket. It is shown, that the dual brackets and dual hamiltonians can be obtained from the canonical (PB) and from the initial hamiltonian by imposing of the second kind constraints on the initial dynamical system, on the closed string model in the constant background fields, as example. The closed string model in the constant background fields is considered without constraints, with the second kind constraints and with first kind constraints as the B-chiral string. The two particles discrete closed string model is considered as two relativistic particle system to show the difference between the Gupta-Bleuler method of the quantization with the first kind constraints and the quantization of the Dirac bracket with the second kind constraints.

hep-th↗
arXiv2001

Bihamiltonity as origin of T-duality of the closed string model

In assumption,that string model is the integrable model for particular kind of the background fields, the closed string model in the background gravity field and the antisymmetric B-field is considered as the bihamiltonian system. It is shown, that bihamiltonity is origin of T-duality of the string models. The new Poisson brackets, depending of the background fields and of their derivatives, are obtained. The integrability condition is obtained as the compactability of the bihamoltonity condition and the Jacobi identity of the new Poisson bracket. The B-chiral string model is dual to the chiral string model for the constant background fields.

hep-th↗
arXiv1997

The consistent reduction of the differential calculus on the quantum group $GL_{q}(2,C)$ to the differential calculi on its subgroups and $σ$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C{q}(2|0)$ and infinitesimal transformations

Explicit construction of the second order left differential calculi on the quantum group and its subgroups are obtained with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the 3-dimensional differential calculi on the quantum subgroup $SL_q(2,C)$, the differential calculi on the Borel subgroups $B_{L}^{(2)}(C)$, $B_{U}^{(2)}(C)$ of the lower and of the upper triangular matrices, on the quantum subgroups $U_{q}(2)$, $SU_{q}(2)$, $Sp_{q}(2,C)$, $Sp_{q}(2)$, $T_{q}(2,C)$, $B_{L}(C)$, $B_{U}(C)$, $U_{q}(1)$, $Z_{-}^{(2)}(C)$, $Z_{+}^{(2)}(C)$ and on the their real forms. The classical limit ($q\to 1$) of the left differential calculus is the nondeformed differential calculus. The differential calculi on the Borel subgroups $B_{L}(C)$, $B_{U}(C)$ of the $SL_{q}(2,C)$ coincide with two solutions of Wess-Zumino differential calculus on the quantum plane $C_q(2|0)$. The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$ quantum group symmetry over two-dimensional nondeformed Minkovski space and in the $σ$-models with ${SL_{q}(2,R)/U_{p}(1)}$, $C_{q}(2|0)$ quantum group symmetry is considered. The Lagrangian formalism over the quantum group manifolds is discussed. The variational calculus on the $SL_{q}(2,R)$ group manifold is obtained. The classical solution of $C_{q}(2|0)$ {$σ$}-model is obtained.

q-alg↗
arXiv1997

$σ$-models on the quantum group manifolds $SL_{q}(2,R)$, $SL_{q}(2,R)/U_{h}(1)$, $C_{q}(2|0)$ and infinitesimal trasformations

The differential and variational calculus on the $SL_{q}(2,R)$ group is constructed. The spontaneous breaking symmetry in the WZNW model with $SL_{q}(2,R)$ quantum group symmetry and in the $σ$-models with ${SL_{q}(2,R)/U_{h}(1)}$ ,$C_{q}(2|0)$ quantum group symmetry is considered. The Lagrangian formalism over the quantum group manifolds is discussed. The classical solution of $C_{q}(2|0)$ {$σ$}-model is obtained.

q-alg↗
arXiv1995

Matched differential calculus on the quantum groups $GL_q(2,C),SL_q(2,C),C_q(2|0)$

We proposed the construction of the differential calculus on the quantum group and its subgroup with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the differential calculus on the quantum subgroup $SL_q(2,C)$ and quantum plane $C_q(2|0)$ (''quantum matrjoshka''). We found, that there are two differential calculi, associated to the left differential Maurer--Cartan 1-forms and to the right differential 1-forms. Matched reduction take the degeneracy between the left and right differentials. The classical limit ($q\to 1$) of the ''left'' differential calculus and of the ''right'' differential calculus is undeformed differential calculus. The condition ${\cal D}_qG=1$ gives the differential calculus on $SL_q(2,C)$, which contains the differential calculus on the quantum plane $C_q(2|0)$.

q-alg↗