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Renat Gontsov

Publications and source records attributed to Renat Gontsov.

13 recordsLinked to original sources
arXiv2024

Convergence of (generalized) power series solutions of functional equations

Solutions of nonlinear functional equations are generally not expressed as a finite number of combinations and compositions of elementary and known special functions. One of the approaches to study them is, firstly, to find formal solutions (that is, series whose terms are described and ordered in some way but which do not converge apriori) and, secondly, to study the convergence or summability of these formal solutions (the existence and uniqueness of actual solutions with the given asymptotic expansion in a certain domain). In this paper we deal only with the convergence of formal functional series having the form of an infinite sum of power functions with (complex, in general) power exponents and satisfying analytical functional equations of the following three types: a differential, $q$-difference or Mahler equation.

math.CA↗
arXiv2021

From formal to actual Puiseux series solutions of algebraic differential equations of first order

The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated Painleve theorem cannot be applied explicitly for the study of convergence. Several examples illustrating relationships to the Painleve theorem and lesser-known Petrovic's results are provided.

math.CA↗
arXiv2021

Triangular Schlesinger systems and superelliptic curves

We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size $(p\times p)$ are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference $q$, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference $q$, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the $(2\times2)$-case, we obtain explicit sequences of rational solutions and one-parameter families of rational solutions of Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.

math-ph↗
arXiv2020

On the convergence of generalized power series solutions of $q$-difference equations

A sufficient condition for the convergence of a generalized formal power series solution to an algebraic $q$-difference equation is provided. The main result leans on a geometric property related to the semi-group of (complex) power exponents of such a series. This property corresponds to the situation in which the small divisors phenomenon does not arise. Some examples illustrating the cases where the obtained sufficient condition can be or cannot be applied are also depicted.

math.CA↗
arXiv2019

On the convergence of exotic formal series solutions of an ODE. A proof by the implicit mapping theorem

We propose a sufficient condition of the convergence of a complex power type formal series of the form $φ=\sum_{k=1}^{\infty}α_k(x^{{\rm i}γ})\,x^k$, where $α_k$ are functions meromorphic at the origin and $γ\in{\mathbb R}\setminus\{0\}$, that satisfies an analytic ordinary differential equation (ODE) of a general type. An example of a such type formal solution of the third Painlevé equation is presented and the proposed sufficient condition is applied to check its convergence.

math.CA↗
arXiv2017

On the convergence of formal Dulac series satisfying an algebraic ODE

We propose a sufficient condition of the convergence of a Dulac series formally satisfying an algebraic ordinary differential equation (ODE). Such formal solutions of algebraic ODEs appear rather often, in particular, the third, fifth, and sixth Painlevé equations possess formal Dulac series solutions, whose convergence follows from the proposed sufficient condition.

math.CA↗
arXiv2013

Solvability of linear differential systems in the Liouvillian sense

The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular ones and propose some criteria of solvability for systems whose (formal) exponents are sufficiently small.

math.CA↗