The discharge shock wave in a condensed ideally conducting medium with a magnetic field

Using a model equation of state containing a point of inflection, we solve the problem of the formation of a discharge shock wave in an ideally conducting medium with a magnetic field. Dissipative processes are neglected. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 51–53.     

On high-temperature superconductivity

We consider electron-phonon interaction models for the case of strong coupling as well as the well-known Hubbard model. We write out the asymptotics as the number of electrons tends to infinity; this allows us to determine the temperature of the superconducting-to-normal state phase transition and the jump in the heat capacity. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 706–722, November, 1999.     

Magnetic catalysis in a P-even, chiral-invariant three-dimensional model with four-fermion interaction

We study how an external magnetic field H affects the phase structure of a P-even, chiral-invariant three-dimensional model of field theory with a four-fermion interaction. An arbitrarily small, but nonzero, magnetic field induces the spontaneous breaking of the initial symmetry (the magnetic catalysis phenomenon). Depending on the values of the coupling constants, the model vacuum is either P-even or chiral invariant for H≠0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 323–338, August, 2000.     

The effective action and the spectrum of collective modes in the antiferromagnetic phase of the three-band repulsive hubbard model

The paper is concerned with the application of path integration (successive integration over “fast” and “slow” variables and the Hubbard-Stratonovich transformation) to the antiferromagnetic state in the three-band, two-dimensional repulsive Hubbard model. It is shown that, in the lowest approximation, the model under consideration is equivalent to the half-filled single-band Hubbard model. The effective action for the collective Bose-fields, the critical temperature of the phase transition, and the spectra of excitations are obtained. Bibliography: 24 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 216–230. Translated by C. L. Malyshev.     

Effective fermion models near a tricritical point

We consider the class of fermion models with a momentum cutoff Λ and an interaction containing derivatives. For the quasilocal fermion model with two coupling constants near a tricritical point, we classify the solutions of mass-gap equations for different values of coupling constants. The mass spectrum of the scalar excitations in the vicinity of the tricritical point is derived and it is found that two scalar exitations always arise when at least one of the constants exceeds its critical value. The mass formation for scalar bound states induced by static gluon fields (gluon condensates) is investigated. Bibliography 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 5–19. Translated by V. A. Andrianov.     

Electronic and magnetic states in the two-band Hubbard model

Electronic and magnetic states are considered in the two-band Hubbard model. Bibliography:5 titles. Dedicated to the memory of V. N. Popov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 224, 1995, pp. 240–249. Translated by B. Bekker.     

On one method of finding the equilibrium state

An algorithm for solving a linear “production-exchange” model is described. The model reduces to a parametric model with strictly concave utility functions. Some properties of the parametric model are studied. Since the excessive demand map satisfies the discovered preference condition, it is possible to apply the second form of Chebyshev centers. For a sufficiently small parameter, the algorithm converges to the equilibrium state of the initial linear model. Bibliography:11 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 80, 1996, pp. 59–67.     

The existence theorem and exact solutions of a variational problem on high-temperature phase transition with zero surface tension

A model of a two-phase elastic medium with classical energy density is considered. In the case under consideration the energy functional depends on the temperature, which is assumed to be constant along the whole body. The question on the equilibrium state for σ=0 is studied, and a family of exact solutions to the problem is constructed. Bibliography: 8 titles. Translated fromProblemy Matematicheskogo Analiza. No. 15, 1995, pp. 201–212.     

Statistical hydrodynamics of magnetic fluids. I. the nonequilibrium statistical operator method

The Zubarev nonequilibrium statistical operator is used to describe the generalized hydrodynamic state of a magnetic fluid in an external magnetic field. The magnetic fluid is modeled with “liquid-state” and “magnetic” subsystems described using the classical and quantum statistics methods respectively. Equations of the generalized statistical hydrodynamics for a magnetic fluid in a nonhomogeneous external magnetic field with the Heisenberg spin interaction are derived for “liquid-state” and “magnetic” subsystems characterized by different nonequilibrium temperatures. These equations can be used to describe both the weakly and strongly nonequilibrium states. Some limiting cases are analyzed in which the variables of one of the subsystems can be formally neglected. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 132–153, April, 1998.     

Variational problem of the two-phase medium elasticity theory for the zero surface tension coefficient

In this article, we give a proof of the existence theorem for an equilibrium state for the surface tension coefficient σ=0 and investigate the behavior of the equilibrium state for small σ. Bibliography: 4 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1900, pp. 208–225. Translated by S. Yu. Pilyugin.     

On the discrete spectrum of a given SO(2) symmetry of multiparticle systems in potential and homogeneous magnetic fields

For a system of n identical particles in a homogeneous magnetic field, the discrete spectrum of the Hamiltonian H^{α, m} on the subspaces of functions with permutational symmetry α and rotational (SO(2)) symmetry m is studied as m→∞. It is proved that the discrete spectrum of the operator H^α,m contains only one eigenvalue if certain conditions are satisfied. The asymptotic behavior of this eigenvalue as m→∞ is found. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 28–41, 1992. Translated by A. V. Lyakhovskaya     

Remark on the dissipation of the magnetic helicity integral

We prove that the current helicity integral for a thin magnetic tube is directly proportional to the twist Tw of field lines, which appears in the Călugărianu formula as a summand. Using this approach, we analyze the law of the variation of magnetic helicity in a conducting medium with a nonzero magnetic diffusion coefficient and also the equation of the phase transition of the magnetic field under early universe conditions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 150–160, January, 2009.     

Influence of viscosity on the structure of shock waves in the MKdV model

A “non-self-adjoint” integrable MKdV model with boundary conditions of “step-like” type is considered. The time-asymptotic behavior of the solution for the Cauchy problem is obtained. A similar problem for the dissipative perturbation of this problem is discussed. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 199, 1992, pp. 37–42. Translated by R. F. Bikbaev.     

Essential self-adjointness of a discrete magnetic Schrödinger operator

We prove essential self-adjointness for a semibounded from below discrete magnetic Schr?dinger operator in a space that represents a combinatorial model of the two-dimensional Euclidean space. The Dezin discretization scheme is used for constructing a discrete model. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 74–81, January–March, 2008.     

Extensions of dynamical systems and the martingale approximation method

Let T be a measure-preserving transformation of a probability space (X, F, μ) and let A be the generator of a μ-symmetric Markov process with state space X. Under the assumption that A is an “eigenvector” for T an extension of T is constructed in terms of A. By means of this extension a version of the central limit theorem is proved via approximation by martingales. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 10–19. Translated by V. Sudakov.     

Discrete symmetries,Darboux transformation,and exact solutions of the Wess–Zumino–Novikov–Witten model

The matrix Darboux transformation is applied to an auxiliary problem of the classical Wess–Zumino–Novikov–Witten model. One- and two-soliton solutions are written explicitly, and a matrix expression for the N-soliton solution is given. Discrete symmetries of the WZNW model are analyzed, and a solution of the linearized equation of motion is obtained. Bibliography: 19 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 139–152.     

Determination of scalar meson masses in QCD-inspired quark models

We compare two QCD-inspired quark models with four-fermion interaction (with and without the remnant coupling to low-energy gluons) in the regime of dynamical chiral symmetry breaking (DCSB). The first one, the Nambu-Jona-Lasinio (NJL) model, ensures the factorization of scalar and pseudoscalar meson poles in correlators, the well-known Nambu relation between the scalar meson mass and the dynamical quark mass, m_σ=2m_dyn, and the residual chiral symmetry in coupling constants characteristic for the linear σ-model. The second one, the gauge NJL model (GNJL), happens to be qualitatively different from the NJL model, namely, the Nambu relation is not valid, and the factorization of light meson poles does not entail the residual chiral symmetry, i.e., it does not result in a linear σ-model. The more complicated DCSB pattern in the GNJL model is fully explained in terms of excited meson states with the same quantum numbers. The asymptotic restrictions on parameters of scalar and pseudoscalar meson states are derived from the requirement of chiral symmetry restoration at high energies. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 245, 1997, pp. 5–21.     

Critical sets and two-dimensional topologically mixing attractors of a pair of coupled “Tent maps”

Endomorphisms of the plane generated by coupling two identical one-dimensional unimodal piece-wise linear maps are proved to have two-dimensional topologically mixing attractors. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 654–658, May, 1999.     

Four-vertex model

An exactly solvable four-vertex model on a square grid with different boundary conditions is considered. The application of the Algebraic Bethe Ansatz method allows us to calculate the partition function of the model. For fixed boundary conditions, we establish a relation between the scalar product of state vectors with the generating function of column-and row-strict boxed plane partitions. A tiling model on a periodic grid is discussed. Bibliography: 33 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 34–55.     

Mathematical models of marine magnetotelluric sounding

A mathematical model of magnetotelluric sounding (MTS) is constructed for a layered medium with a nonhomogeneous inclusion. The model is intended for marine studies. The problem is solved by the integral electric current method. A modified integral current method is proposed capable of generating a solution with a small number of partitions of the nohomogeneity region. The efficiency of the proposed method is demonstrated for H-polarized field. The behavior of apparent resistivity and magnetic parameter curves is investigated for marine MTS. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 19–28, 2008.