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Том 192, № 2 (2017)
- Год: 2017
- Статей: 11
- URL: https://journals.rcsi.science/0040-5779/issue/view/10435
Article
Kulish–Sklyanin-type models: Integrability and reductions
Аннотация
We start with a Riemann–Hilbert problem (RHP) related toBD.I-type symmetric spaces SO(2r + 1)/S(O(2r − 2s+1) ⊗ O(2s)), s ≥ 1. We consider two RHPs: the first is formulated on the real axis R in the complex-λ plane; the second, on R ⊗ iR. The first RHP for s = 1 allows solving the Kulish–Sklyanin (KS) model; the second RHP is related to a new type of KS model. We consider an important example of nontrivial deep reductions of the KS model and show its effect on the scattering matrix. In particular, we obtain new two-component nonlinear Schrödinger equations. Finally, using the Wronski relations, we show that the inverse scattering method for KS models can be understood as generalized Fourier transforms. We thus find a way to characterize all the fundamental properties of KS models including the hierarchy of equations and the hierarchy of their Hamiltonian structures.
1097-1114
1115-1128
Remark on the reflection coefficient in the Liouville model
Аннотация
We show that the reflection coefficients in the quantum theory of the Liouville model calculated in the bootstrap and Hamiltonian approaches differ from each other by a phase factor and simply yield different normalizations of vertex operators.
1129-1133
1134-1140
Asymmetric six-vertex model and the classical Ruijsenaars–Schneider system of particles
Аннотация
We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric (hyperbolic) functions.
1141-1153
Second-order evaluations of orthogonal and symplectic Yangians
Аннотация
Orthogonal or symplectic Yangians are defined by the Yang–Baxter RLL relation involving the fundamental R-matrix with the corresponding so(n) or sp(2m) symmetry. We investigate the second-order solution conditions, where the expansion of L(u) in u−1is truncated at the second power, and we derive the relations for the two nontrivial terms in L(u).
1154-1161
Quantization of the Kadomtsev–Petviashvili equation
Аннотация
We propose a quantization of the Kadomtsev–Petviashvili equation on a cylinder equivalent to an infinite system of nonrelativistic one-dimensional bosons with the masses m = 1, 2,.... The Hamiltonian is Galilei-invariant and includes the split and merge terms\(\Psi _{{m_1}}^\dag \Psi _{{m_2}}^\dag {\Psi _{{m_1} + {m_2}}}\)and\(\Psi _{{m_1} + {m_2}}^\dag {\Psi _{{m_1}}}{\Psi _{{m_2}}}\)for all combinations of particles with masses m1, m2, and m1 + m2for a special choice of coupling constants. We construct the Bethe eigenfunctions for the model and verify the consistency of the coordinate Bethe ansatz and hence the quantum integrability of the model up to the mass M=8 sector.
1162-1183
Generalized Weyl modules for twisted current algebras
Аннотация
We introduce the notion of generalized Weyl modules for twisted current algebras. We study their representation-theoretic and combinatorial properties and also their connection with nonsymmetric Macdonald polynomials. As an application, we compute the dimension of the classical Weyl modules in the remaining unknown case.
1184-1204
Regularization of Mickelsson generators for nonexceptional quantum groups
Аннотация
Let g′ ⊂ g be a pair of Lie algebras of either symplectic or orthogonal infinitesimal endomorphisms of the complex vector spaces CN−2 ⊂ CNand Uq(g′) ⊂ Uq(g) be a pair of quantum groups with a triangular decompositionUq(g) = Uq(g-)Uq(g+)Uq(h). LetZq(g, g′) be the corresponding step algebra. We assume that its generators are rational trigonometric functions h ∗ → Uq(g±). We describe their regularization such that the resulting generators do not vanish for any choice of the weight.
1205-1217
Rings of h-deformed differential operators
Аннотация
We describe the center of the ring Diffh(n) ofh-deformed differential operators of type A. We establish an isomorphism between certain localizations of Diffh(n) and the Weyl algebra Wn, extended by n indeterminates.
1218-1229
A new generalized Wick theorem in conformal field theory
Аннотация
We describe a new generalized Wick theorem for interacting fields in two-dimensional conformal field theory and briefly discuss its relation to the Borcherds identity and its derivation by an analytic method. We give examples of calculating operator product expansions using the generalized Wick theorem including fermionic fields.
1230-1241