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## Chapter 7. Elliptic and Periodic Asymptotic Forms of Solutions to P5

P. 53-65.

Bruno A. D., Parusnikova A.

We apply methods from Space Power Geometry to the fifth Painlevé equation. Near infinity we obtained 2 families of elliptic asymptotic forms and 4 families of periodic asymptotic forms of its solutions. All of these families are 2-parameter.

Parusnikova A., / Cornell University. Series math "arxiv.org". 2014.

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 these construction to the first five Painleve equations. The seventh section of this work contains results on convergence of formal power series solutions to the fifth Painleve equation near ...

Added: May 11, 2014

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

V.S.Samovol, Mathematical notes 2014 Vol. 95 No. 6 P. 843-855

The paper contains the results of the study of the asymptotic proprties of solutions with integer-valued asymptotics as wel as of solutions arising from the rapid decrease of the coefficient of the equation. To analize the asymptotic behavior of solutions of the equations, methods of power geometry are used. ...

Added: January 25, 2015

Zabrodin A., Zotov A., Journal of Mathematical Physics 2012 Vol. 53 No. 7 P. 073508-1-073508-19

This paper is a continuation of our previous paper where the Painlevé-Calogero correspondence has been extended to auxiliary linear problems associated with Painlevé equations. We have proved, for the first five equations from the Painlevé list, that one of the linear problems can be recast in the form of the non-stationary Schrödinger equation whose Hamiltonian ...

Added: September 19, 2012

V.S. Samovol, Doklady Mathematics 2018 Vol. 97 No. 3 P. 250-253

A Riccati equation with coefficients expandable into convergent power series in a neighborhood of
infinity is considered. Continuable solutions to equations of this type are studied. Conditions for the expansion
of these solutions into convergent series in a neighborhood of infinity are obtained by methods of power
geometry. ...

Added: July 6, 2018

Parusnikova A., / Cornell University Library. 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

Berlin : De Gruyter, 2012

http://www.degruyter.com/view/books/9783110275667/9783110275667.v/9783110275667.v.xml ...

Added: February 16, 2013

Bobrova I., Sokolov V., / Cornell University. 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

Parusnikova A., , in : Banach Center Publications. Vol. 97: Formal and Analytic Solutions of Differential and Difference Equations,.: Warsz. : Polish Academy of Sciences, 2012. 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. ...

Added: March 24, 2013

Parusnikova A., , in : Painlevé Equations and Related Topics. : Berlin : De Gruyter, 2012. Ch. 5. P. 33-38.

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 complex parameters of the equations. They form 16 and 30 families in the neighborhoods of singularpoints z=\infty and z=0, respectively. There are 10 families ...

Added: March 23, 2014

Bruno A. D., Parusnikova A.V., Доклады Академии наук 2012 Vol. 85 No. 1 P. 87-92

By applying methods of power geometry, we find all asymptotic expansions of solutions to the fifth Painlevé equation near its nonsingular point for all values of its four complex parameters. More specifically, 10
families of expansions of solutions to the equation areobtained, of which one was not previously known.
Three expansions are Laurent series, while the remaining ...

Added: March 25, 2014

Bruno A., Parusnikova A., Доклады Академии наук 2011 Т. 438 № 4 С. 439-443

In this work, the methods of power geometry are used to find asymptotic expansions of solutions to the fifth Painlevй equation as x 0 for all values of its four complex parameters. We obtain 30 families of expansions, of which 22 are obtained from published expansions of solutions to the sixth Painlevй equation. Among the ...

Added: April 12, 2012

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

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 30, 2012

Parusnikova A., / Cornell University. 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

Parusnikova A., , in : International Conference “Painlevґe Equations and Related Topics”. : St. Petersburg : The Euler International Mathematical Institute, 2011. 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 = ...

Added: April 16, 2012