A. V. Kitaev

On Boutroux's Tritronquée solutions of the first Painlevé equation

The triply truncated solutions of the first Painleve equation were specified by Boutroux in his famous paper of 1913 as those having no poles (of large modulus) except in one sector of angle 2π/5. There are five such solutions and each of them can be obtained from any other one by applying a certain symmetry transformation. One of these solutions is real on the real axis. We found a characteristic property of this solution, different from the asymptotic description given by Boutroux. This allows us to estimate numerically the position of its real pole and zero closest to the origin. We also study properties of asymptotic series for truncated solutions.

Connection formulae for asymptotics of the fifth Painlevé transcendent on the real axis

In this work we collect together our connection results for asymptotics as t→ + 0 and t→ + ∞ of general and some special solutions of the fifth Painleve equation. Keeping in mind already known and possible applications, actually, a slightly more general object is studied, namely, a nonlinear system of ordinary differential equations (ODEs) governing isomonodromy deformations for the (2×2) matrix linear ODE, .

Connection formulae for the first Painlevé transcendent in the complex domain

We have found, via the isomonodromy deformation method, a complete asymptotic description of the first Painlevé transcendent in the complex domain.

Turning Points of Linear Systems and Double Asymptotics of the Painlevé Transcendents

Recently, on the basis of the isomonodromy deformation method (IDM) [1], a considerable success has been achieved in the description of asymptotic properties of the Painleve transcendents especially in obtaining so-called connection formulae [2, 3, 4, 5], Moreover, there is no problem in using the same IDM technique for obtaining asymptotic formulae of Boutroux type in the complex domain, if the asymptotic behavior of a solution is given on a possible special ray (the analogs of Stokes rays) [6, 7]. It is also possible to obtain the justification of IDM asymptotic formulae by intrinsic means of the IDM. For different approaches to this problem, we refer to [8, 9, 10].

On the linearization of the first and second Painlevé equations

We found Fuchs–Garnier pairs in 3 × 3 matrices for the first and second Painleve equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve equation we use the generalized Laplace transform to derive an invertible integral transformation relating two of its Fuchs–Garnier pairs in 2 × 2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and that found by Harnad, Tracy and Widom. Together with the certain other transformations it allows us to relate all known 2 × 2 matrix Fuchs–Garnier pairs for the second Painleve equation to the original Garnier pair.

An isomonodromy cluster of two regular singularities

We consider a linear 2 × 2 matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the ODE. In particular, a zero-distance limit for the ODE exists. The monodromy group of the limiting ODE is calculated in terms of the original one. This coalescing process generates a limit for the corresponding nonlinear systems of isomonodromy deformations. In our main example the latter limit reads as P 6 → P 5 , where P n is the nth Painleve equation. We also discuss some general problems which arise while studying the above-mentioned limits for the Painleve equations.

Non-Schlesinger deformations of ordinary differential equations with rational coefficients

We consider deformations of 2×2 and 3×3 matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which do not satisfy the well known system of Schlesinger equations (or its natural generalization). Some general statements concerning the reducibility of such deformations for 2×2 ODEs are proved. An explicit example of the general non-Schlesinger deformation of 2×2-matrix ODEs of the Fuchsian type with four singular points is constructed and application of such deformations to the construction of special solutions of the corresponding Schlesinger systems is discussed. Some examples of isomonodromic and non-isomonodromic deformations of 3×3 matrix ODEs are considered. The latter arise as the compatibility conditions with linear ODEs with non-single-valued coefficients.

Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

Algebraic solutions of the sixth Painlevé equation can be constructed with the help of RS-transformations of hypergeometric equations. Construction of these transformations includes specially ramified rational coverings of the Riemann sphere and the corresponding Schlesinger transformations (S-transformations). Some algebraic solutions can be constructed from rational coverings alone, without obtaining the corresponding pullbacked isomonodromy Fuchsian system, i.e., without the S part of the RS transformations. At the same time, one and the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painlevé VI solutions. In the case of high degree coverings, construction of the S parts of the RS-transformations may represent some computational difficulties. This paper presents computation of explicit RS pullback transformations and derivation of algebraic Painlevé VI solutions from them. As an example, we present a computation of all seed solutions for pullbacks of hyperbolic hypergeometric equations. Bibliography: 26 titles.

Fluctuations near the Boundaries in the Six-Vertex Model

Horizontal-arrow fluctuations near the boundaries in the six-vertex model with domain-wall boundary conditions are considered. For these fluctuations, a representation in terms of the standard objects of the theory of orthogonal polynomials is obtained. This representation is used for the study of the large N limit. Bibliography: 18 titles.