Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Polynomial Solutions of Fractional Polynomial Differential Equations Generated by the Second Painleve Equation

Author(s):

Zilya Nailevna Khakimova

Mozhaisky Military Space Academy

vka@mail.ru

Larisa Nikolaevna Timofeeva

Mozhaisky Military Space Academy

vka@mail.ru

Oleg Valentinovich Zaitsev

BSTU «VOENMEH» named after D.F. Ustinov

zaytsev_oleg1997@mail.ru

Abstract:

A discrete pseudogroup of point, Bäcklund and tangent transformations for the 2nd Painlevé equation, closed in classes of ordinary differential equations with polynomial or fractional polynomial right-hand side, is constructed. A 36-vertex graph of this pseudogroup is constructed. The solutions in polynomials of all polynomial and fractional polynomial equations of the orbit of the 2nd Painlevé equation are calculated for some integer values of the parameter of the right-hand side. In addition, a method is indicated for finding solutions of the orbital equations of the 2nd coefficients on the right-hand side.

Keywords

• discrete group
• ordinary differential equation (ODE)
• Painlevé transcendents
• second Painlevé equation
• transformation pseudogroup

References:

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