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Asymptotic expansions of solutions to the fifth Painlevé equation in neighbourhoods of singular and nonsingular points of the equation
P. 113–124.
Applying methods of plane Power Geometry we are looking for the asymptotic expansions of solutions to the fifth Painleve ́ equation in the neighbourhood of its singular and nonsingular points.
In book
Vol. 97: Formal and Analytic Solutions of Differential and Difference Equations,. , Warsz.: Polish Academy of Sciences, 2012.
N.V. Gaianov, A.V. Parusnikova, Siberian Mathematical Journal 2024 Vol. 65 No. 5 P. 1062–1073
We derive a special form of the q-difference equations whose solutions exist in the form of a Dulac series. We find the coefficients of the Dulac series from some algebraic difference equations having a polynomial solution under appropriate conditions. An example is given of a q-difference equation which demonstrates the lack of upper bounds for the degrees of ...
Added: September 25, 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
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
Alfimov G. L., Fedotov A. P., Sinelshchikov D., Physica D: Nonlinear Phenomena 2020 Vol. 402 Article 132245
We present a method for determination of the blow up point for a complex nonautonomous ordinary differential equation with a cubic nonlinearity. This equation describes stationary nonlinear modes for two important NLS-based models: (i) the Gross–Pitaevskii equation with complex potential and repulsive nonlinearity and (ii) the Lugiato–Lefever equation in the case of normal dispersion. We ...
Added: November 8, 2019
Parusnikova A., Vasilyev A., Journal of Mathematical Sciences 2019 Vol. 241 No. 3 P. 318–326
We examine asymptotic expansions of the third Painlevé transcendents for αδ ≠ 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic ...
Added: October 26, 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
V.S.Samovol, Mathematical notes 2019 Vol. 105 No. 4 P. 592–603
The Riccati equation with coefficients expandable in convergent power series in a neigh- borhood of infinity are considered. Extendable solutions of such equations are studied. Methods of power geometry are used to obtain conditions for convergent series expansions of these solutions. An algorithm for deriving such series is given. ...
Added: April 10, 2019
Bershtein M., Shchechkin A., / Series math "arxiv.org". 2018.
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation equals to the series of c=1 Virasoro conformal blocks. We study similar series of c=−2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blow-up relations on Nekrasov partition functions.
We also study series of q-deformed c=−2 ...
Added: November 22, 2018
Parusnikova A., Vasilyev A. V., / Series arXiv "math". 2017. No. 1702.05758.
In this paper we present a family of values of the parameters of the third Painlevé equation such that Puiseux series formally satisfying this equation -- considered as series of z^{2/3} -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them ...
Added: February 21, 2017
Parusnikova A., Васильев А. В., Итоги науки и техники. Современная математика и ее приложения. Тематические обзоры 2017 Т. 139 С. 70–78
Проведено асимптотическое исследование третьих трансцендентов Пенлеве при α δ ≠ 0, γ = 0 в окрестности бесконечности в некотором секторе с углом раствора < 2 π методом доминантных мономов (англ. Method of dominant balance). Промежуточные результаты сравниваются с результатами, полученными при использовании методов трехмерной степенной геометрии. Найдены возможные асимптотики, выраженные в терминах эллиптических функций, а ...
Added: February 21, 2017
Васильев А. В., Parusnikova A., В кн.: Дифференциальные уравнения и смежные вопросы математики.Труды VIII Приокской научной конференции.: Государственный социально-гуманитарный университет, 2016. С. 34–43.
В данной работе для поиска асимптотик решений третьего уравнения Пенлеве в окрестности бесконечности применяются метод доминантных мономов и трехмерная степенная геометрия. ...
Added: February 21, 2017