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Irina Bobrova

Publications and source records attributed to Irina Bobrova.

12 recordsLinked to original sources
arXiv2026

A non-commutative discrete first Painlev\'e hierarchy: the Lax pair approach

Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlev\'e hierarchy, we construct a non-commutative version of this hierarchy, denoted by $\text{d-PI}_m^{\text{nc}}$. We show that both hierarchies, $\text{d-PI}_m$ and $\text{d-PI}_m^{\text{nc}}$, can be expressed in terms of the polynomials $S_s^k(n)$, which we call the Svinin polynomials. We also derive a reduction of the non-commutative Volterra lattice hierarchy to the $\text{d-PI}_m^{\text{nc}}$ hierarchy and present explicit continuous limits for the first three members of the $\text{d-PI}_m^{\text{nc}}$, thereby recovering non-commutative analogues of the first three members of the differential first Painlev\'e hierarchy.

nlin.SI↗
arXiv2026

On a non-commutative sixth $q$-Painlevé system: from discrete system to surface theory

In this paper, we describe the non-commutative formal geometry underlying a certain class of discrete integrable systems. Our main example is a non-commutative analog, labeled $q$-P$(A_3)$, of the sixth $q$-Painlevé equation. The system $q$-P$(A_3)$ is constructed by postulating an extended birational representation of the extended affine Weyl group $\widetilde{W}$ of type $D_5^{(1)}$ and by selecting the same translation element in $\widetilde{W}$ as in the commutative case. Starting from this non-commutative discrete system, we develop a non-commutative version of Sakai$'$s surface theory, which allows us to derive the same birational representation that we initially postulated. Moreover, we recover the well-known cascade of multiplicative discrete Painlevé equations rooted in $q$-P$(A_3)$ and establish a connection between $q$-P$(A_3)$ and the non-commutative $d$-Painlevé systems introduced in I. Bobrova. Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlevé equations.

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arXiv2024

Non-Abelian discrete Toda chains and related lattices

We have derived a non-abelian analog for the two-dimensional discrete Toda lattice which possesses solutions in terms of quasideterminants and admits Lax pairs of different forms. Its connection with non-abelian analogs for several well-known (1+1) and one-dimensional lattices is discussed. In particular, we consider a non-commutative analog of the scheme: discrete Toda equations $\rightarrow$ Somos-$N$ sequences $\rightarrow$ discrete Painlevé equations.

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arXiv2024

Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlev\'e equations

A non-abelian generalisation of a birational representation of affine Weyl groups and their application to the discrete dynamical systems is presented. By using this generalisation, non-commutative analogs for the discrete systems of $A_n^{(1)}$, $n \geq 2$ type and of $d$-Painlev\'e equations with an additive dynamic were derived. A coalescence cascade of the later is also discussed.

nlin.SI↗
arXiv2023

On symmetries of the non-stationary PIIn hierarchy and their applications

In the current paper we study auto-Bäcklund transformations of the non-stationary second Painlevé hierarchy $\text{P}_\text{II}^{(n)}$ depending on $n$ parameters: a parameter $α_n$ and times $t_1, \dots, t_{n-1}$. Using generators $s^{(n)}$ and $r^{(n)}$ of these symmetries, we have constructed an affine Weyl group $W^{(n)}$ and its extension $\tilde{W}^{(n)}$ associated with the $n$-th member considered hierarchy. We determined $\text{P}_\text{II}^{(n)}$ rational solutions via Yablonskii-Vorobiev-type polynomials $u_m^{(n)} (z)$. We brought out a correlation between Yablonskii-Vorobiev-type polynomials and polynomial $τ$-functions $τ_m^{(n)} (z)$ and found their determinant representation in the Jacobi-Trudi form.

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arXiv2023

Classification of Hamiltonian non-abelian Painlevé type systems

All Hamiltonian non-abelian Painlevé systems of ${\rm{P}}_{1}-{\rm{P}}_{6}$ type with constant coefficients are found. For ${\rm{P}}_{1}-{\rm{P}}_{5}$ systems, we replace an appropriate inessential constant parameter with a non-abelian constant. To prove the integrability of new ${\rm{P}}_{3}^{\prime}$ and ${\rm{P}}_{5}$ systems thus obtained, we find isomonodromic Lax pairs for them.

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arXiv2023

Different linearizations of non-abelian second Painlevé systems and related monodromy surfaces

In this paper, we discuss a connection between different linearizations for non-abelian analogs of the second Painlevé equation. For each of the analogs, we listed the pairs of the Harnard-Tracy-Widom (HTW), Flaschka-Newell (FN), and Jimbo-Miwa (JM) types. A method for establishing the HTW-JM correspondence is suggested. For one of the non-abelian analogs, we derive the corresponding non-abelian generalizations of the monodromy surfaces related to the FN- and JM-type linearizations. A natural Poisson structure associated with these monodromy surfaces is also discussed.

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arXiv2023

On classification of non-abelian Painlevé type systems

We find all non-abelian generalizations of $\text{P}_1 - \text{P}_6$ Painlevé systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax representations.

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arXiv2022

Non-abelian Painlevé systems with generalized Okamoto integral

We study non-abelian systems of Painlevé type. To derive them, we introduce an auxiliary autonomous system with the frozen independent variable and postulate its integrability in the sense of the existence of a non-abelian first integral that generalizes the Okamoto Hamiltonian. All non-abelian P6-P2-systems with such integrals are found. A coalescence limiting scheme is constructed for these non-abelian Painlevé systems. This allows us to construct an isomonodromic Lax pair for each of them.

nlin.SI↗
arXiv2020

The Sigma Form for the PII Hierarchy

In this paper we study the so-called sigma form of the second Painlevé hierarchy. To obtain this form, we use some properties of the Hamiltonian structure of the second Painlevé hierarchy and of the Lenard operator.

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