Asymptotic description of solutions of the fourth Painlevé equations on the stokes rays analogs

The proof of the existence of classical solutions is given for the Painlevé type nonlinear ordinary differential equations. The solutions have asymptotic formulas, which can be obtained by the isomonodromic deformation method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 101–109, 1989.     

Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions

In 1991, one of the authors showed the existence of quadratic transformations between the Painlevé VI equations with local monodromy differences (1/2, a, b, ±1/2) and (a, a, b, b). In the present paper we give concise forms of these transformations. They are related to the quadratic transformations obtained by Manin and Ramani–Grammaticos–Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic‐pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painlevé VI functions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Determination of trap parameters by simultaneous measurement of thermostimulated luminescence and thermostimulated conductivity

A method is proposed for processing of thermostimulated luminescence (TSL) curve in the presence of thermostimulated conductivity (TSC) in which no assumption is made as to the smallness of dn_c/dt (n_c is the conduction electron concentration). Results of the theory are applied to study of ZnSAg, Al and ZnSCu single crystals, in which elementary traps are revealed, and their activation energy and the value of the frequency factor determined. For ZnSCu the radiative transition recombination section is calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 94–101, June, 1977.     

The isomonodromy approach to matric models in 2D quantum gravity

We consider the double-scaling limit in the hermitian matrix model for 2D quantum gravity associated with the measure exp . We show that after the appropriate modification of the contour of integration the Cross-Migdal-Douglas-Shenker limit to the Painlevé I equation (in the generic case of the pure gravity) is valid and calculate the nonperturbative parameters of the corresponding Painlevé function. Our approach is based on the WKB-analysis of the L-A pair corresponding to the discrete string equation in the framework of the Inverse Monodromy Method. Here we extend our results, which were obtained before for the particular casesN=2,3. Our analysis complements the isomonodromy approach proposed by G. Moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself.     

Ferromagnetic and antiferromagnetic ordering in the wurtzite-type diluted magnetic semiconductors

We determined the magnetic symmetry groups (Shubnikov groups) of the wurtzite-type crystals doped with magnetic atoms periodically distributed at cation sites. The magnetic groups of doped crystals with ferro- or antiferromagnetic properties arise from the P6₃ mc (C _6v ⁴), P3m1 (C _3v ¹), C _s ³ (Cm) or C ₁¹ (P1) ordinary space groups. Those arising from the P6₃ mc or P3m1 group can present macroscopic ferromagnetic polarization only along the wurtzite symmetry axis whereas those arising from the C _s ³ (Cm) group can present ferromagnetic polarization only in the symmetry plane and those arising from the C ₁¹ (P1) group can present polarization along any direction. Actual structures do not keep in general long-range order for magnetic atom distribution but the samples with macroscopic polarization should present local symmetries close to those listed above. Therefore, orientation of easy magnetization can provide information about the most frequent local symmetry.     

An Entire Function Defined by a Nonlinear Recurrence Relation

For the nonlinear recurrence relation it is proved that the limit exists and defines an entire function of ₂ = ₁(1-₁).     

Turning points of linear systems and the double asymptotics of the Painlevé transcendents

On the basis of the connection between the theories of linear and nonlinear special functions, we present a method which makes it possible to consider the well known formal limits from complicated Painlevé equations to simpler ones as the double asymptotics of specific solutions of these equations with respect to the parameter and the argument under some special relation between them. The hierarchies of the first and second Painlevé equations are interpreted as special functions that describe the isomonodromic collision of turning points for linear systems of ordinary differential equations. Bibliography: 28 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 187, pp. 53–74, 1990. Translated by B. M. Bekker.