Alexander V. Kitaev

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Grothendieck’s dessins d’enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of isomonodromy type. It is shown that higher-order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Bely̆ı functions. Moreover, deformations of(More)
We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d’enfants related with the construction of the algebraic solutions of the sixth Painlevé equation via the method of RS-transformations. 2000 Mathematics Subject Classification: 34M55, 33E17, 33E30(More)
We introduce a new notion, a special function of the isomonodromy type, and show that most of the functions known in applied mathematics and mathematical physics as special functions belong to this type. This deenition provides a uniied approach to the theories of \linear" special functions, i.e., classical higher transcen-dental functions, and \non-linear"(More)
Vertical-arrow fluctuations near the boundaries in the six-vertex model on the two-dimensional NxN 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(More)
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(More)
We discuss relations which exist between analytic functions belonging to the recently introduced class of special functions of the isomonodromy type (SFITs). These relations can be obtained by application of some simple transformations to auxiliary ODEs with respect to a spectral parameter which associated with each SFIT. We consider two applications of(More)
Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leading-order asymptotics as t → ±∞ of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation,(More)
We consider deformations of 2×2 and 3×3 matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which don’t satisfy the wellknown system of Schlesinger equations (or its natural generalization). Some general statements concerning reducibility of such deformations for 2× 2 ODEs are proved. An explicit example of the(More)
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 RamaniGrammaticos-Tamizhmani via(More)