arXiv subjects

Julia Bernatska

Publications and source records attributed to Julia Bernatska.

25 records · Page 2Linked to original sources

arXiv2014

Orbit Approach to Separation of Variables in sl(3)-Related Integrable Systems

Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{sl}(3)$ loop algebra. We consider two types of generic orbits embedded into a common manifold, endowed with two nonsingular Lie-Poisson brackets. We prove that separation of variables on orbits of both types is realized by the same variables of separation. We also construct the integrable systems on these orbits: a coupled 3-component nonlinear Schrödinger equation and an isotropic SU(3) Landau-Lifshitz equation.

nlin.SI↗

arXiv2013

SU(3) magnet: finite-gap integration on the lowest genus curve

We consider the integrable system of isotropic SU(3) Landau-Lifshits equation as a Hamiltonian system on a coadjoint orbit of the SU(3) loop group. We connect the mentioned equation with an isotropic SU(3) magnet because it describes the mean fields of magnetic and quadrupole moments in a spin-1 lattice. For the system of isotropic SU(3) Landau-Lifshits equation we perform separation of variables in Sklyanin's manner, and integrate in the lowest finite gap where the spectral curve is elliptic.

nlin.SI↗

arXiv2013

Harmonic analysis on Lagrangian manifolds of integrable Hamiltonian systems

For an integrable Hamiltonian system we construct a representation of the phase space symmetry algebra over the space of functions on a Lagrangian manifold. The representation is a result of the canonical quantization of the integrable system in terms of separation variables. The variables are chosen in such way that a half of them parameterizes the Lagrangian manifold, which coincides with the Liouville torus of the integrable system. The obtained representation is indecomposable and non-exponentiated.

math-ph↗

arXiv2010

Ordered States and Nonlinear Large-Scale Excitations in a Plane Magnet with Spin s=1

We study ordered states and topological excitations in a quasi-two-dimensional magnet, modeled by a square lattice with spins $s {=} 1$ at all sites, and the Hamiltonian with biquadratic exchange interaction between nearest neighbor sites. We propose two effective Hamiltonians for description of large-scale excitations in the two-dimensional case. They describe excitations of the mean field in a nematic phase and a mixed ferromagnetic-nematic phase. It is shown that the effective Hamiltonians are minimized on configurations with fixed topological charge. These topological excitations can arise at low temperatures and cause a destruction of a long-range order in the two-dimensional system.

cond-mat.str-el↗

arXiv2010

Topological excitations in 2D spin system with high spin $s>= 1$

We construct a class of topological excitations of a mean field in a two-dimensional spin system represented by a quantum Heisenberg model with high powers of exchange interaction. The quantum model is associated with a classical one (the continuous classical analogue) that is based on a Landau-Lifshitz like equation, and describes large-scale fluctuations of the mean field. On the other hand, the classical model is a Hamiltonian system on a coadjoint orbit of the unitary group SU($2s {+} 1$) in the case of spin $s$. We have found a class of mean field configurations that can be interpreted as topological excitations, because they have fixed topological charges. Such excitations change their shapes and grow preserving an energy.

cond-mat.mes-hall↗

arXiv2010

On Separation of Variables for Integrable Equations of Soliton Type

We propose a general scheme for separation of variables in the integrable Hamiltonian systems on orbits of the loop algebra $\mathfrak{sl}(2,\Complex)\times \mathcal{P}(λ,λ^{-1})$. In particular, we illustrate the scheme by application to modified Korteweg--de Vries (MKdV), sin(sinh)-Gordon, nonlinear Schrödinger, and Heisenberg magnetic equations.

nlin.SI↗

arXiv2008

Geometry and Topology of Coadjoint Orbits of Semisimple Lie Groups

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were the subject of much study. However, they remain hard for calculation and application. We propose simple solutions for the following problems: an explicit parameterization of the orbit by means of a generalized stereographic projection, obtaining a Kählerian structure on the orbit, introducing basis two-forms for the cohomology group of the orbit.

math.RT↗