Asymptotic expansions of solutions of the sixth Painlevé equation

Abstract

Abstract. We obtain all asymptotic expansions of solutions of the sixth Painlevé equation near all three singular points x = 0, x = 1, and x = ∞ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable x. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point x = 0 for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones.

Application of Power Geometry and Normal Form Methods to the Study of Nonlinear ODEs

“…For these purposes we use the condition A. Since it would take a lot of space, and because it was presented earlier in several papers (see [2,3]) we do not repeat it here. The condition A was formulated and applied to the considered problem in [4].…”

“…A method of analysis of the integrability of the system (1) based on Power Transformations [1] and computation of normal forms near stationary solutions of transformed systems (see [2] and Ch. II in [3]) was proposed in [4,5].…”

Expansions of solutions to the fifth Painlevé equation near its nonsingular point

“…Equation (1) has two sin gular points: z = 0 and z = ∞. By applying methods of two dimensional power geometry [1,2], we find all asymptotic expansions of solutions (AES) to Eq. (1) near its nonsingular point z = z 0 , z 0 ≠ 0, z 0 ≠ ∞ for all values of the parameters α, β, γ, and δ.…”

“…Multiplying it by (t + z 0 ) 2 w(w -1) and moving all its terms to the right hand side yields (2) Our goal is to find AES of Eq. (2) as t → 0.…”

Power geometry and expansions of solutions to the Painlevé equations

“…I consider this question for three Painlevé equations P 3 , P 5 and P 6 , because among 6 Painlevé equations P 1 -P 6 there are 3 equations P 3 , P 5 , P 6 having complicated and exotic expansions of solutions ( [4][5][6]). First coefficients for equations P 3 , P 5 and P 6 are polynomials in log x in complicated expansions and usual or Laurent polynomials in x iγ in exotic expansions [4,6]. Each of the Painlevé equations P 3 , P 5 and P 6 has 4 complex parameters a, b, c, d. Two of them are included into the truncated equation.…”