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A. V. Kitaev

Publications and source records attributed to A. V. Kitaev.

At least 19 recordsLinked to original sources
arXiv2026

The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point

We give a classification for the small-$τ$ asymptotic behaviours of solutions to the degenerate third Painlevé equation, $u^{''}(τ) = \frac{(u^{\prime}(τ))^{2}}{u(τ)} - \frac{u^{\prime}(τ)}τ + \frac{1}τ\left(-8 \varepsilon (u(τ))^{2} + 2ab \right) + \frac{b^{2}}{u(τ)}, \quad\varepsilon=\pm1,\quad\varepsilon b>0, \quad a\in\mathbb{C}\setminus i\mathbb{Z}$, in terms of the monodromy data of a $2\times2$ matrix linear ODE whose isomonodromy deformations they describe. We also study the complete asymptotic expansions of the solutions.

math.CA↗
arXiv2025

Painlevé Property and Generating Functions for Asymptotics

This paper proposes a new approach to the asymptotic analysis of Painlevé equations. The approach is based on representing solutions of the Painlevé equations using formal series in two variables, $\sum_{k=0}^{\infty}y^kA_k(x)$, with rational functions $A_k(x)$. The approach is applied to the asymptotic analysis of the third degenerate Painlevé equation.

math.CA↗
arXiv2023

One-Parameter Meromorphic Solution of the Degenerate Third Painlevé Equation with Formal Monodromy Parameter $a=\pm i/2$ Vanishing at the Origin

We prove that there exists a one-parameter meromorphic solution $u(τ)$ vanishing at $τ=0$ of the degenerate third Painlevé equation, \begin{equation*} u^{\prime \prime}(τ) \! = \! \frac{(u^{\prime}(τ))^{2}}{u(τ)} \! - \! \frac{u^{\prime}(τ)}τ \! + \! \frac{1}τ \! \left(-8 \varepsilon (u(τ))^{2} \! + \! 2ab \right) \! + \! \frac{b^{2}}{u(τ)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0, \end{equation*} for formal monodromy parameter $a=\pm i/2$. We study number-theoretic properties of the coefficients of the Taylor-series expansion of $u(τ)$ at $τ=0$ and its asymptotic behaviour as $τ\to+\infty$. These asymptotics are visualized for generic initial data.

math.CA↗
arXiv2023

Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter

Various properties of algebroid solutions of the degenerate third Painlevé equation, \begin{equation*} u^{\prime \prime}(τ) \! = \! \frac{(u^{\prime}(τ))^{2}}{u(τ)} \! - \! \frac{u^{\prime}(τ)}τ \! + \! \frac{1}τ \! \left(-8 \varepsilon (u(τ))^{2} \! + \! 2ab \right) \! + \! \frac{b^{2}}{u(τ)},\qquad \varepsilon=\pm1,\quad\varepsilon b>0, \end{equation*} for the monodromy parameter $a=0$ are studied. The paper contains connection results for asymptotics as $τ\to+0$ and as $τ\to+\infty$ for $a\in\mathbb{C}$. Using these results, the simplest algebroid solution with asymptotics $u(τ)\to cτ^{1/3}$ as $τ\to0$, where $c\in\mathbb{C}\setminus\{0\}$, together with its associated integral $\int_0^τ{(u(t))^{-1}\,d t}$, are considered in detail, and their basic asymptotic behaviours are visualized.

math.CA↗
arXiv2019

Connection Formulae for Asymptotics of the Fifth Painlevé Transcendent on the Imaginary Axis: I

Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlevé equation as $t\to\imath\infty$ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE. $$ \frac{d}{dλ}Y= \left(\frac t2σ_3 + \frac{A_0}λ+\frac{A_1}{λ-1}\right)Y. $$ The parametrization allows one to derive connection formulas for the asymptotics. We provide numerical verification of the results. Important special cases of the connection formulas are also considered.

math.CA↗
arXiv2010

Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II

The degenerate third Painleve' equation, $u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$, where $c=+/-1$, $b>0$, and $a$ is a complex parameter, is studied via the Isomonodromy Deformation Method. Asymptotics of general regular and singular solutions $u(t)$ as $t -> +/-\infty$ and $t -> +/-i\infty$ are derived and parametrized in terms of the monodromy data of the associated 2X2 linear auxiliary problem introduced in the first part of this work [1]. Using these results, three-real-parameter families of solutions that have infinite sequences of zeroes and poles that are asymptotically located along the real and imaginary axes are distinguished: asymptotics of these zeroes and poles are also obtained.

math.CA↗
arXiv2008

On the Linearization of the First and Second Painleve' Equations

We found Fuchs--Garnier pairs in 3X3 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 its Fuchs--Garnier pairs in 2X2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with the certain other transformations it allows us to relate all known 2X2 matrix Fuchs--Garnier pairs for the second Painleve' equation with the original Garnier pair.

math.CA↗
arXiv2007

On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System

We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3x3 matrix pairs, we found an integral auto-transformation for the standard Fuchs--Garnier pair for the fifth Painleve' equation. It generates an Okamoto-like Bäcklund transformation for the fifth Painleve' equation. Another application is an integral transformation relating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve' equation.

math.CA↗
arXiv2007

Boundary Conditions for Scaled Random Matrix Ensembles in the Bulk of the Spectrum

A spectral average which generalises the local spacing distribution of the eigenvalues of random $ N\times N $ hermitian matrices in the bulk of their spectrum as $ N\to\infty $ is known to be a $τ$-function of the fifth Painlevé system. This $τ$-function, $ τ(s) $, has generic parameters and is transcendental but is characterised by particular boundary conditions about the singular point $s=0$, which we determine here. When the average reduces to the local spacing distribution we find that $τ$-function is of the separatrix, or partially truncated type.

math.CA↗
arXiv2006

An Isomonodromy Cluster of Two Regular Singularities

We consider a linear $2\times2$ 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 $P_6\to P_5$, where $P_n$ is the $n$-th Painlevé equation. We also discuss some general problems which arise while studying the above-mentioned limits for the Painlevé equations.

math.CA↗
arXiv2004

The Dirichlet Boundary Value Problem for Real Solutions of the first Painlevé Equation on Segments on Non-Positive Semi-Axis

We develop a qualitative theory for real solutions of the equation $y''=6y^2 -x$. In this work a restriction $x\leq0$ is assumed. An important ingredient of our theory is the introduction of several new transcendental functions of one, two, and three variables that describe different properties of the solutions. In particular, the results obtained allow us to completely analyse the Dirichlet boundary value problem $y(a)=y^0$, $y(b)=y_0$ for $a<b\leq0$.

math.CA↗
arXiv2003

Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painlevé Equation. I

The degenerate third Painlevé equation, $u^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}τ + \frac{1}τ(-8 εu^{2} + 2ab) + \frac{b^{2}}{u}$, where $ε,b \in \mathbb{R}$, and $a \in \mathbb{C}$, and the associated tau-function are studied via the Isomonodromy Deformation Method. Connection formulae for asymptotics of the general as $τ\to \pm 0$ and $\pm i0$ solution and general regular as $τ\to \pm \infty$ and $\pm i \infty$ solution are obtained.

math.CA↗
arXiv2003

Dessins d'Enfants, Their Deformations and Algebraic the Sixth Painlevé and Gauss Hypergeometric Functions

We consider an application of Grothendieck's dessins d'enfants to the theory of the sixth Painlevé and Gauss hypergeometric functions: two classical special functions of the isomonodromy type. It is shown that, higher order transformations and the Schwarz table for the Gauss hypergeometric function are closely related with some particular Belyi functions. Moreover, we introduce a notion of deformation of the dessins d'enfants and show that one dimensional deformations are a useful tool for construction of algebraic the sixth Painlevé functions.

nlin.SI↗
arXiv2002

Boundary polarization in the six-vertex model

Vertical-arrow fluctuations near the boundaries in the six-vertex model on the two-dimensional $N \times N$ square lattice with the domain wall boundary conditions are considered. The one-point correlation function (`boundary polarization') is expressed via the partition function of the model on a sublattice. The partition function is represented in terms of standard objects in the theory of orthogonal polynomials. This representation is used to study the large N limit: the presence of the boundary affects the macroscopic quantities of the model even in this limit. The logarithmic terms obtained are compared with predictions from conformal field theory.

cond-mat.stat-mech↗
arXiv2001

Transformations ${RS}_4^2(3)$ of the Ranks $\leq4$ and Algebraic Solutions of the Sixth Painlevé Equation

Compositions of rational transformations of independent variables of linear matrix ordinary differential equations (ODEs) with the Schlesinger transformations ($RS$-transformations) are used to construct algebraic solutions of the sixth Painlevé equation. $RS$-Transformations of the ranks 3 and 4 of $2\times2$ matrix Fuchsian ODEs with 3 singular points into analogous ODE with 4 singular points are classified.

nlin.SI↗