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Book chapter

Chapter 5. Asymptotic expansions of solutions to the fifth Painlevé equation

P. 33-38.
Parusnikova A.

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 in the neighborhood of a nonsingular point. Over 20 families are new.

Language: English
Full text (PDF, 2.45 Mb)
Keywords: Painleve equations

In book

Chapter 5. Asymptotic expansions of solutions to the fifth Painlevé equation
Painlevé Equations and Related Topics
Berlin: De Gruyter, 2012.
Similar publications
Chapter 7. Elliptic and Periodic Asymptotic Forms of Solutions to P5
Bruno A. D., Parusnikova A. In bk.: Painlevé Equations and Related Topics. Berlin: De Gruyter, 2012. P. 53-65.

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.  

Added: Feb 16, 2013
On Gevrey orders of power expansions of solutions to the third and fifth Painleve equations
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: Oct 20, 2013
Локальные разложения решений пятого уравнения Пенлеве
Брюно А. Д., Парусникова А. В. 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 other eight families, one was previously known and two can be obtained from the expansions of solutions to the third Painlevй equation. Three families of half-exotic expansions and two families of complicated expansions are new.

Added: Apr 12, 2012
Разложения решений пятого уравнения Пенлеве в окрестности его неособой точки
Брюно А. Д., Парусникова А. В. Доклады Академии наук. 2012. Т. 442. № 5. С. 583-588.
Added: Nov 30, 2012
Asymptotic expansions of solutions to the fifth Painlevé equation in neighbourhoods of singular and nonsingular points of the equation
Parusnikova A. In bk.: 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: Mar 24, 2013
4-Dimensional Power Geometry and its Application to P1 - P5
Parusnikova A. Working papers by Cornell University. 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: Mar 28, 2015
Quantum Painlevé-Calogero correspondence for Painlevé VI
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 is a natural quantization of the classical Calogero-like Hamiltonian for the corresponding Painlevé equation. In the present paper we establish the quantum Painlevé-Calogero correspondence for the most general case, the Painlevé VI equation. We also show how the desired special gauge and the needed choice of variables can be derived starting from the corresponding Schlesinger system with rational spectral parameter.

Added: Sep 19, 2012
Asymptotic expansions of solutions to the fifth Painleve equation
Parusnikova A. In bk.: 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 = 0 correspondingly. There exist 10 families in the neighbourhood of nonsingular point. Over 20 families are new.

Added: Apr 16, 2012
Painlevé Equations and Related Topics
Berlin: De Gruyter, 2012.

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

Added: Feb 16, 2013