Quantization scheme for modular <Emphasis Type="Italic">q</Emphasis>-difference equations

We consider modular pairs of certain second-order q-difference equations. An example of such a pair is the t-Q Baxter equations for the quantum relativistic Toda lattice in the strong coupling regime. Another example from quantum mechanics is q-deformation of the Schrödinger equation with a hyperbolic potential. We show that the analyticity condition for the wave function or the Baxter function leads to a set of transcendental equations for the coefficients of the potential or the transfer matrix, the solution of which is their discrete spectrum.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 500–509, March, 2005.     

Stability properties of cycles and tori of a simplest nonresonant wave-type equation

For a number of meaningful examples, nonresonant wave-type equations are shown to be characterized by periodic dynamics. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 744–750, November, 1997. Translated by O. V. Sipacheva     

Basic theonean triangles

Properties of the set of triangles with integer sides whose area is the square of an integer are studied. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 561–569, April, 1997. Translated by V. N. Dubrovsky     

Mean value theorems for solutions of linear partial differential equations

We consider generalized mean value theorems for solutions of linear differential equations with constant coefficients and zero right-hand side which satisfy the following homogeneity condition with respect to a given vectorM with positive integer components: for each partial derivative occurring in the equation, the inner product of the vector composed of the orders of this derivative in each variable by the vectorM is independent of the derivative. The main results of this paper generalize the well-known Zalcman theorem. Some corollaries are given.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 260–272, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01366.     

On a transformation operator

We prove the existence of a transformation operator that takes the solution of the equationy″=λ²ⁿ y to the solution of the equation

Gauge-invariant discrete models of Yang-Mills equations

Two discrete models of Yang-Mill equations are constructed in the space ℝⁿ for some matrix-valued Lie group. A gauge-invariant discrete model is examined. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 742–754, May, 1997. Translated by O. V. Sipacheva     

A discrete and q asymptotic iteration method

ABSTRACT

We introduce a finite difference and q-difference analogues of the Asymptotic Iteration Method of Ciftci, Hall, and Saad. We give necessary, and sufficient condition for the existence of a polynomial solution to a general linear second-order difference or q-difference equation subject to a ‘terminating condition’, which is precisely defined. When a difference or q-difference equation has a polynomial solution, we show how to find the second solution.     

Exact solutions of the spherically symmetric gravitational field equations

The Einstein gravitational equations in the spherically symmetric case and for the dust model (i.e., p = 0) have been studied by several authors. However, the solutions obtained by them are not completed yet, and the corresponding metric is written in implicit forms which is inconvenient for physical problems. In the present paper we make the following improvements: (1) We obtain all spherical solutions for the dust model with explicit expressions which consist of three classes and an exceptional case; (2) All these solutions contain singularities which are analyzed together with their physical properties. __________ Translated from Journal of Fudan University (Natural Science), 1974, 1: 92–98 An erratum to this article is available at .     

Existence of periodic solutions

An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 769–784, May, 1997. Translated by V. N. Dubrovsky     

Tricomi problem for strongly degenerate equations of parabolic-hyperbolic type

A new integral representation of solutions of a Tricomi problem for a strongly degenerate system of equations of parabolic-hyperbolic type is constructed. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp.385–392, September, 1999.     

Convergence of periodically forced rank-type equations

Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves, this is analogous to a differential equation with periodic forcing. A large class of such non-autonomous difference equations are shown to converge to a periodic limit, which is independent of the initial condition. The period of the limit does not depend on how far back each term is allowed to look back in the sequence, and is in fact equal to the period of the forcing.     

On solvability conditions of the Dirichlet problem form-curvature equations

This papor contains a refined variant of the Jenkins-Serrin-Trudinger condition related to the Dirichlet problem for curvature equations. The new condition implies the classical solvability of the Gauss curvature equation in the case of positive curvature small enough on the boundary. Bibliography: 8 titles. Dedicated to V. A. Solonnikov on his sixtieth anniversary Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 48–65. Translated by S. A. Titarenko.     

An initial boundary-value problem for a system of equations of magnetohydrodynamics

In the present paper we consider one initial boundary-value problem for a system of equations of magnetohydrodynamics in the case where it is necessary to take into account the displacement currents in the Maxwell system of equations. We prove a local (in time) unique solvability of this problem in the Sobolev spaces. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 228–254, April–June, 2000. Translated by R. Lapinskas     

The exact order of information-based complexity of weakly singular integral equations with periodic analytic coefficients

The exact order of complexity of weakly singular integral equations including logarithmic singularities with periodic and analytic coefficients is found. This class of equations contains the boundary equations of exterior boundary value problems for the two-dimensional Helmholtz equation. Translated fromMatematicheskie Zarnetki, Vol. 62, No. 5, pp. 643–656, November, 1997. Translated by O. V. Sipacheva     

The kernel of the neumann operator for a strictly diffractive analytic problem

We consider p a partial differential operator of order 2 and Rⁿ= ω⁺ ∪ ?ω ∪ ω^? a partition of Rⁿ , such that (p, ω⁺) admits a strictly diffractive point (in the sense of Friedlander and Melrose). We compute the trace and the trace of the normal derivative on ?ω of the solution u of the diffraction problem pu= 0 in ω⁺ u satisfying a mixed boundary condition on ?ω, ?ω analytic. That is done using the construction by Lebeau of a Gevrey 3 parametrix in the neighborhood of the strictly diffractive point.

This result generalizes, for a mixed boundary condition, the Gevrey 3 propogation result of Lebeau. We use this result to compute the leading term in the shadow region of the diffracted wave outside a strictly convex analytical obstacle with a mixed boundary condition and a given incoming wave.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

A nonlinear problem for the Laplace equation with a degenerating Robin condition

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution.     

The Boundary Regularity of a Weak Solution of the Navier-Stokes Equation and its Connection to the Interior Regularity of Pressure

We assume that v is a weak solution to the non-steady Navier-Stokes initial-boundary value problem that satisfies the strong energy inequality in its domain and the Prodi-Serrin integrability condition in the neighborhood of the boundary. We show the consequences for the regularity of v near the boundary and the connection with the interior regularity of an associated pressure and the time derivative of v.     

Saddle-point method and resurgent analysis

The topological part of the theory of the parameter-dependent Laplace integral is known to consist of two stages. At the first stage, the integration contour is reduced to a sum of paths of steepest descent for some value of the parameter. At the second stage, this decomposition (and hence the asymptotic expansion of the integral) is continued to all other parameter values. In the present paper, the second stage is studied with the help of resurgent analysis techniques. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 278–296, February, 1997. Translated by B. Yu. Sternin and V. E. Shatalov