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Asymptotic expansions of solutions to the fifth Painleve equation
P. 126–131.
By means of Power Geometry we obtained all asymptotic expansions of solutions to the equation P5 of the following five types: power, power-logarithmic, complicated, exotic and half-exotic for all values of 4 complex parameters of the equation. They form 16 and 30 families in the neighbourhood of singular points z = infty and z = 0 correspondingly. There exist 10 families in the neighbourhood of nonsingular point. Over 20 families are new.
In book
St. Petersburg: The Euler International Mathematical Institute, 2011.
Gaianov N., Parusnikova A., Уфимский математический журнал 2026 Т. 18 № 2 С. 14–22
We consider an algebraic 𝑞–difference equation. We propose a sufficient condition for the existence of a formal power–logarithmic expansion in the vicinity of zero of the solution to such an equation. We apply this sufficient condition to construct the formal expansion of a solution to a certain 𝑞–difference analogue of the fifth Painlevé equation for
particular ...
Added: June 24, 2026
Gaianov N., Parusnikova A., / Cornell University. Серия math "arxiv.org". 2025.
An algebraic q-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this sufficient condition for constructing a formal expansion of a solution to a certain q-difference analogue of the fifth Painlevé equation ...
Added: December 25, 2025
Бляхман Л.Г., Громов Е.М., ISBN 978-5-907868-27-4, 2024.
Приводится банк задач контрольных работ по математическому анализу для студентов международного бакалавриата по бизнесу и экономики. С примерами решений нулевых выариантов. ...
Added: May 17, 2024
I. A. Bobrova, Sokolov V. V., Journal of Geometry and Physics 2023 Vol. 191 Article 104885
We find all non-abelian generalizations of P1 - P6 Painleve systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax representations. ...
Added: June 21, 2023
Bobrova I., Sokolov V., Journal of Nonlinear Mathematical Physics 2023 Vol. 30 No. 2 P. 646–662
All Hamiltonian non-abelian Painlevé systems of P1−P6 type with constant coefficients are found. For P1−P5 systems, we replace an appropriate inessential constant parameter with a non-abelian constant. To prove the integrability of new P′3 and P5 systems thus obtained, we find isomonodromic Lax pairs for them. ...
Added: December 23, 2022
Bibilo Y., Glutsyuk A., Nonlinearity 2022 Vol. 35 No. 10 P. 5427–5480
The tunnelling effect predicted by Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modelled by a family of differential equations on two-torus depending on three parameters: B (abscissa), A (ordinate), ω ...
Added: December 20, 2022
Bobrova I., Теоретическая и математическая физика 2022 Т. 213 № 1 С. 65–94
We study auto-Bäcklund transformations of the nonstationary second Painlevé hierarchy $\rm{P}_{\rm{II}}^{(n)}$ depending on n parameters: a parameter $\alpha_n$ and times $t_1, …, t_{n−1}$. Using generators $s^{(n)}$ and $r^{(n)}$ of these symmetries, we construct an affine Weyl group $W^{(n)}$ and its extension $\tilde{W}^{(n)}$ associated with the nth member of the hierarchy. We determine rational solutions of $\rm{P}_{\rm{II}}^{(n)}$ in terms of Yablonskii–Vorobiev-type polynomials $u_m^{(n)} (z)$. We show that Yablonskii–Vorobiev-type polynomials are related to the polynomial τ-function $\tau_m^{(n)} (z)$ and find their determinant representation ...
Added: October 8, 2022
Lomonosov T., , in: Abstracts of the 9th International Conference on Differential and Functional Differential Equations.: [б.и.], 2022. P. 77–78.
An algebraic approach to solving certain nonhomogeneous ODEs is studied. The approach is also extendable to solving certain nonhomogeneous difference equations. The approach is based on representing the right-hand side of the equation as an element of a certain vector space, the restriction of the d/dx operator on which is an automorphism space. ...
Added: July 5, 2022
Bobrova I., Sokolov V., / Series arXiv "math". 2022.
We study non-abelian systems of Painleve type. To derive them, we introduce anauxiliary 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 ...
Added: June 22, 2022
Аношин В. И., Бекетова А. Д., Parusnikova A., В кн.: Дифференциальные уравнения и смежные вопросы математики. Труды XIII Приокской научной конференции.: Государственный социально-гуманитарный университет, 2021. С. 33–39.
Added: March 28, 2022
Anoshin V. I., Beketova A., Parusnikova A. et al., Programming and Computer Software 2022 Vol. 48 No. 1 P. 30–35
Asymptotic behavior and asymptotic expansions of solutions to the second term of the fourth Painlevé hierarchy are constructed using power geometry methods [1]. Only results for the case of general position—for the equation parameters β,δ≠0β,δ≠0—are provided. For constructing asymptotic expansions, a code written in a computer algebra system is used. ...
Added: February 5, 2022
Bobrova I., Mazzocco M., Journal of Geometry and Physics 2021 Vol. 166 Article 104271
In this paper we study the so-called sigma form of the second Painleve hierarchy. To obtain this form, we use some properties of the Hamiltonian structure of the second Painleve hierarchy and of the Lenard operator. ...
Added: September 25, 2021
Springer, 2020.
This edited volume gathers selected, peer-reviewed contributions presented at the fourth International Conference on Differential & Difference Equations Applications (ICDDEA), which was held in Lisbon, Portugal, in July 2019.
First organized in 2011, the ICDDEA conferences bring together mathematicians from various countries in order to promote cooperation in the field, with a particular focus on applications. ...
Added: November 1, 2020
Kondratieva L. A., A.V. Romanov, Electronic Journal of Qualitative Theory of Differential Equations 2019 No. 96 P. 1–11
We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in Rn permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincaré–Bendixson theory. In the case n = 3 we implement such a scenario for a model of a satellite rotation around a celestial body of ...
Added: December 22, 2019
Rabinowitch A., Московский технологический университет (МИРЭА), 2018.
В учебно-методическом пособии рассмотрены основные методы решения обыкновенных дифференциальных уравнений. Даются общие подходы к решению линейных и нелинейных дифференциальных уравнений первого порядка. Подробно излагаются основные методы решения линейных дифференциальных уравнений порядка выше первого и систем линейных дифференциальных уравнений. Рассматриваются также численные методы решения обыкновенных дифференциальных уравнений. Приводятся многочисленные примеры решения рассмотриваемых классов дифференциальных уравнений и ...
Added: October 5, 2019
Parusnikova A., Vasilyev A. V., Journal of Dynamical and Control Systems 2019 Vol. 25 No. 4 P. 681–690
In this paper, we study the third Painlevé equation with parameters γ = 0, αδ ≠ 0. The Puiseux series formally satisfying this equation (after a certain change of variables) asymptotically approximate of Gevrey order one solutions to this equation in sectors with vertices at infinity. We present a family of values of the parameters δ = −β^2/2 ≠ 0 such that ...
Added: June 4, 2019
Гурарий М. М., Жаров М. М., Русаков С. Г. et al., Наноиндустрия 2018 № 82 С. 410–411
The paper considers the problems of envelope following method construction for determining transient response and steady state of integrated circuits. The envelope following algorithms based on using one-step high order integration methods have been offered. ...
Added: February 12, 2019