Article
Classification of Hamiltonian Non-Abelian Painlevé Type Systems
Journal of Nonlinear Mathematical Physics. 2023. Vol. 30. No. 2. P. 646–662.
Bobrova I., Sokolov V.
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.
Levin A., Ольшанецкий М. А., Зотов А. В., Успехи математических наук 2014 Т. 69 № 1(415) С. 39–124.
В данной работе изомонодромные задачи описываются в терминах плоских G-расслоений на проколотых эллиптических кривых Σ_τ и связностей с регулярными особенностями в отмеченных точках. Расслоения классифицируются по их характеристическим классам, которые являются элементами группы вторых когомологий H^2(Σ_τ,Z(G)), где Z(G) – центр G. По каждой простой комплексной группе Ли G и произвольному характеристическому классу определяется пространство модулей ...
Added: January 21, 2015
Anoshin V. I., Beketova A., Parusnikova A. et al., Computational Mathematics and Mathematical Physics 2023 Vol. 63 No. 1 P. 86–95.
The second member of the fourth Painlevé hierarchy is considered. Convergence of certain power asymptotic expansions in a neighborhood of zero is proved. New families of power asymptotic expansions are found. Computations are carried out using a computer algebra system. Reference to a code that can be used for computing the Gevrey order of the ...
Added: March 30, 2023
Parusnikova A., / Series "Working papers by Cornell University". 2014. No. 1412.6690..
In the first section of this work we introduce 4-dimensional Power Geometry for second-order ODEs of a polynomial form. In the next five sections we apply this construction to the first five Painlev ́e equations. ...
Added: March 28, 2015
Gavrylenko P., Lisovyy O., / Series math-ph "arXiv". 2016. No. 1608.00958..
We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL(N,ℂ). The corresponding operator acts in the direct sum of N(n−3) copies of L2(S1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of ...
Added: September 20, 2016
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., / ИПМ им. М.В. Келдыша РАН. Серия :: "ИПМ им. М.В. Келдыша РАН". 2013. № 97..
В данной работе рассматривается вопрос о суммируемости по Жевре степенных разложений решений четвёртого уравнения Пенлеве в окрестности бесконечности в случае общего положения αβ ≠ 0. Для анализа используются методы французской и японской школ, алгоритмы сравниваются с алгоритмами степенной геометрии. ...
Added: October 25, 2013
Bershtein M., Shchechkin A., Letters in Mathematical Physics 2019 Vol. 109 No. 11 P. 2359–2402.
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal 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 blowup relations on Nekrasov partition functions. We also study series of ...
Added: August 31, 2020
Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073507-1–073507-19.
The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary ...
Added: September 19, 2012
Parusnikova A., /. 2013. No. 1310.5345..
The question under consideration is Gevrey summability of power expansions of solutions to the third and fifth Painlev\'{e} equations near infinity. Methods of French and Japaneese schools are used to analyse these properties of formal power series solutions. The results obtained are compared with the ones obtained by means of Power Geometry. ...
Added: October 20, 2013
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
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
Levin A., Olshanetsky M., Zotov A., Journal of Physics A: Mathematical and Theoretical 2016 Vol. 49 No. 39 P. 1–24.
In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops, such Lax pairs with spectral parameters follow from the associative Yang–Baxter equation and its ...
Added: September 17, 2016
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., Nonlinearity 2022 Vol. 35 No. 12 P. 6528–6556.
Using the Painleve–Kovalevskaya test, we find several polynomial matrix
systems, which can be regarded as non-commutative generalisations of the
Painleve-4 equation. For these systems isomonodromic Lax pairs are presented.
Limiting transitions that reduce them to known matrix Painleve-2 equations are
found. ...
Added: October 29, 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
I. A. Bobrova, Journal of Mathematical Physics 2023 Vol. 64 No. 10 Article 101702.
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 ...
Added: October 6, 2023
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
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
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
Gavrylenko P., Lisovyy O., / Series arXiv.org "math-ph". 2017. No. 1705.01869..
We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painlev\'e III equation of type $D_8$ (radial sine-Gordon equation). In particular, ...
Added: May 5, 2017
Bruno A., Parusnikova A., Доклады Академии наук 2012 Т. 442 № 5 С. 583–588.
В работе методами степенной геометрии найдены все асимптотические разложения решений пятого уравнения Пенлеве в окрестности его не особой точки для всех значений четырех комплексных параметров уравнения. Получено 10 семейств разложений решений уравнения, одно из которых не было известно раньше. Три разложения являются рядами Лорана, а остальные семь – рядами Тейлора. Все они сходятся в (проколотой) ...
Added: November 29, 2012
Bershtein M., Shchechkin A., Letters in Mathematical Physics 2019 Vol. 109 No. 11 P. 2359–2402.
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal 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 blowup relations on Nekrasov partition functions. We also study series of ...
Added: October 21, 2019
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
Anastasia V. Parusnikova, Opuscula Mathematica 2014 Vol. 34 No. 3 P. 591–599.
The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painlevй equations near infinity. We consider the fifth Painleve equation in two cases: when αβγδ \neq 0 and when αβγ \neq 0, δ = 0 and the third Painlevé equation when all the parameters of the equation are ...
Added: February 28, 2014
Levin A., Ольшанецкий М. А., Зотов А. В., Успехи математических наук 2014 Т. 69 № 1(415) С. 39–124.
В данной работе изомонодромные задачи описываются в терминах плоских G-расслоений на проколотых эллиптических кривых Σ_τ и связностей с регулярными особенностями в отмеченных точках. Расслоения классифицируются по их характеристическим классам, которые являются элементами группы вторых когомологий H^2(Σ_τ,Z(G)), где Z(G) – центр G. По каждой простой комплексной группе Ли G и произвольному характеристическому классу определяется пространство модулей ...
Added: January 21, 2015
Anoshin V. I., Beketova A., Parusnikova A. et al., Computational Mathematics and Mathematical Physics 2023 Vol. 63 No. 1 P. 86–95.
The second member of the fourth Painlevé hierarchy is considered. Convergence of certain power asymptotic expansions in a neighborhood of zero is proved. New families of power asymptotic expansions are found. Computations are carried out using a computer algebra system. Reference to a code that can be used for computing the Gevrey order of the ...
Added: March 30, 2023
Parusnikova A., / Series "Working papers by Cornell University". 2014. No. 1412.6690..
In the first section of this work we introduce 4-dimensional Power Geometry for second-order ODEs of a polynomial form. In the next five sections we apply this construction to the first five Painlev ́e equations. ...
Added: March 28, 2015
Gavrylenko P., Lisovyy O., / Series math-ph "arXiv". 2016. No. 1608.00958..
We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL(N,ℂ). The corresponding operator acts in the direct sum of N(n−3) copies of L2(S1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of ...
Added: September 20, 2016
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., / ИПМ им. М.В. Келдыша РАН. Серия :: "ИПМ им. М.В. Келдыша РАН". 2013. № 97..
В данной работе рассматривается вопрос о суммируемости по Жевре степенных разложений решений четвёртого уравнения Пенлеве в окрестности бесконечности в случае общего положения αβ ≠ 0. Для анализа используются методы французской и японской школ, алгоритмы сравниваются с алгоритмами степенной геометрии. ...
Added: October 25, 2013
Bershtein M., Shchechkin A., Letters in Mathematical Physics 2019 Vol. 109 No. 11 P. 2359–2402.
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal 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 blowup relations on Nekrasov partition functions. We also study series of ...
Added: August 31, 2020
Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073507-1–073507-19.
The Painlevé-Calogero correspondence is extended to auxiliary linear problems associated with Painlevé equations. The linear problems are represented in a new form which has a suggestive interpretation as a "quantized" version of the Painlevé-Calogero correspondence. Namely, the linear problem responsible for the time evolution is brought into the form of non-stationary Schrödinger equation in imaginary ...
Added: September 19, 2012
Parusnikova A., /. 2013. No. 1310.5345..
The question under consideration is Gevrey summability of power expansions of solutions to the third and fifth Painlev\'{e} equations near infinity. Methods of French and Japaneese schools are used to analyse these properties of formal power series solutions. The results obtained are compared with the ones obtained by means of Power Geometry. ...
Added: October 20, 2013
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
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
Levin A., Olshanetsky M., Zotov A., Journal of Physics A: Mathematical and Theoretical 2016 Vol. 49 No. 39 P. 1–24.
In this paper we suggest generalizations of elliptic integrable tops to matrix-valued variables. Our consideration is based on the R-matrix description which provides Lax pairs in terms of quantum and classical R-matrices. First, we prove that for relativistic (and non-relativistic) tops, such Lax pairs with spectral parameters follow from the associative Yang–Baxter equation and its ...
Added: September 17, 2016
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., Nonlinearity 2022 Vol. 35 No. 12 P. 6528–6556.
Using the Painleve–Kovalevskaya test, we find several polynomial matrix
systems, which can be regarded as non-commutative generalisations of the
Painleve-4 equation. For these systems isomonodromic Lax pairs are presented.
Limiting transitions that reduce them to known matrix Painleve-2 equations are
found. ...
Added: October 29, 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
I. A. Bobrova, Journal of Mathematical Physics 2023 Vol. 64 No. 10 Article 101702.
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 ...
Added: October 6, 2023
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
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
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
Gavrylenko P., Lisovyy O., / Series arXiv.org "math-ph". 2017. No. 1705.01869..
We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painlev\'e III equation of type $D_8$ (radial sine-Gordon equation). In particular, ...
Added: May 5, 2017
Bruno A., Parusnikova A., Доклады Академии наук 2012 Т. 442 № 5 С. 583–588.
В работе методами степенной геометрии найдены все асимптотические разложения решений пятого уравнения Пенлеве в окрестности его не особой точки для всех значений четырех комплексных параметров уравнения. Получено 10 семейств разложений решений уравнения, одно из которых не было известно раньше. Три разложения являются рядами Лорана, а остальные семь – рядами Тейлора. Все они сходятся в (проколотой) ...
Added: November 29, 2012
Bershtein M., Shchechkin A., Letters in Mathematical Physics 2019 Vol. 109 No. 11 P. 2359–2402.
Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation is equal 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 blowup relations on Nekrasov partition functions. We also study series of ...
Added: October 21, 2019
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
Anastasia V. Parusnikova, Opuscula Mathematica 2014 Vol. 34 No. 3 P. 591–599.
The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painlevй equations near infinity. We consider the fifth Painleve equation in two cases: when αβγδ \neq 0 and when αβγ \neq 0, δ = 0 and the third Painlevé equation when all the parameters of the equation are ...
Added: February 28, 2014