Quantum corrections to ϕ 4 model solutions and applications to Heisenberg chain dynamics

The Heisenberg spin chain is considered in ? ⁴ model approximation. Quantum corrections to classical solutions of the one-dimensional ? ⁴ model within the correspondent physics are evaluated with account of rest d-1 dimensions of a d-dimensional theory. A quantization of the model is considered in terms of spacetime functional integral. The generalized zeta-function formalism is used to renormalize and evaluate the functional integral and quantum corrections to energy in a quasiclassical approximation. The results are applied to appropriate conditions of the spin chain model and its dynamics, for which elementary solutions, energy and the quantum corrections are calculated.     

Covariance of Lax Pairs and Integrability of the Compatibility Condition

We study the joint covariance of Lax pairs (LPs) with respect to Darboux transformations (DT). The scheme is based on comparing general expressions for the transformed coefficients of a LP and its Frechet derivative. We use the compact expressions of the DT via a version of non-Abelian Bell polynomials. We show that the so-called binary version of Bell polynomials forms a convenient basis for specifying the invariant subspaces. Some nonautonomous generalizations of KdV and Boussinesq equations are discussed in this context. We consider a Zakharov–Shabat-like problem to obtain restrictions at a minimal operator level. The subclasses that allow a DT symmetry (covariance at the LP level) are considered from the standpoint of dressing-chain equations. The cases of the classical DT and binary combinations of elementary DTs are considered with possible reduction constraints of the Mikhailov type (generated by an automorphism). Examples of Liouville–von Neumann equations for the density matrix are considered as illustrations.     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

Propagation of a boundary disturbance in a stratified gas for arbitrary knudsen number

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 70–79, September–October, 1993.     

Dynamics of Domain Walls in a Cylindrical Amorphous Ferromagnetic Microwire with Magnetic Inhomogeneities

Theoretical and Mathematical Physics - We study the dynamics of a domain wall (DW) in the magnetic core of thin amorphous glass-coated bistable microwires with a circular cross section containing...     

Study of free convective boundary layer of isothermal lateral surface of axisymmetrical horizontal body

Approximate analytical solution of simplified Navier–Stokes and Fourier–Kirchhoff equations describing free convective heat transfer from isothermal surface has been presented. It is supposed that the surface has the horizontal axis of symmetry and its axial cross-section lateral boundary is a concave function. The equation for the boundary layer thickness is derived for typical for natural convection assumptions. The most important are that the convective fluid flow is stationary and the normal to the surface component of velocity is negligibly small in comparison with the tangential one. The theoretical results are verified by two characteristic cases of the revolution surfaces namely for horizontal conic and vertical round plate. Both limits of presented solution coincide with known formulas.     

Interaction of Vortical and Acoustic Waves: From General Equations to Integrable Cases

The equations of the (2+1)-dimensional boundary-layer perturbation split into eigenmodes: a vortex wave and two acoustic waves. We assume that the equations of state (Taylor series approximation) are arbitrary. We realize a mode definition via local-relation equations extracted from the linearization of the general system over the boundary-layer flow. Each such link determines an invariant subspace and the corresponding projector. We examine the nonlinear equation for a vortex wave using a special orthogonal coordinate system based on streamlines. The equations for the orthogonal curves are linked to the Laplace equations via Laplace and Moutard transformations. The nonlinearity determines the proper form of the interaction between vortical and acoustic boundary-layer perturbation fields fixed by projecting to a subspace of the Orr-Sommerfeld equation solutions for the Tollmienn-Schlichting (linear vortical) wave and by the corresponding procedure for the acoustic wave. We suggest a new mechanism for controlling the nonlinear resonance of the Tollmienn-Schlichting wave by sound via a four-wave interaction.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 171–181, July, 2005.     

Problem of proper decomposition and initialization of acoustic and entropy modes in a gas affected by the mass force

The relations connecting perturbations specific for acoustic and entropy modes in an accelerated fluid or in a fluid affected by constant mass force, are derived. They allow to decompose the total vector of perturbations and the overall energy into acoustic and non-acoustic parts uniquely at any instant. In order to do this, three quantities are required, for example total perturbations in entropy, pressure and velocity. The evaluations are made in regard to the content of acoustic and non-acoustic parts of the total energy excluding its kinetic part. In some cases, exact relations may be derived.     

A KdV equation in 2+1 dimensions: Painlevé analysis,solutions and similarity reductions

The nonlinear equationm _ty =(m _yxx +m _x m _y ) _x is throughly analyzed. The Painlevé test yields a positive result. The Bäckhand transformations are found and the Darboux-MoutardMatveev formalism arises in the context of this analysis. Some solutions and their interactions are also analyzed. The singular manifold equations are also used to determine symmetry reductions. This procedure can be related with the direct method of Clarkson and Kruskal.