A class of differential equations of PI-type with the quasi-Painlevé property

Abstract

We present a certain class of second order nonlinear differential equations containing the first Painlevé equation (PI). Each equation in it admits the quasi-Painlevé property, namely every movable singularity of a general solution is at most an algebraic branch point. For these equations we show some basic properties.

A Class of Non-Linear ODEs with Movable Algebraic Singularities

“…Proof. The proof is very similar to the corresponding proof in [3], [9] or [10]. The solution of the first-order linear differential equation 10can be found by the method of variation of the constant.…”

“…2. A function W , rational in w and linear in w is constructed which satisfies a first-order linear differential equation of the form (10). Lemma 4 together with Lemma 5 then shows that W is bounded as z → z ∞ alongγ.…”

“…The next step is to allow also for movable singularities with a finite branching. This property is called the quasi-Painlevé property in [10] or also weak-Painlevé property in [8]. Here, we consider rational second-order ODEs of the form (1) y = E(z, y)(y ) 2 + F (z, y)y + G(z, y),…”

“…[5], [7], [9], and subsequently for classes of equations with the quasi-Painlevé property, e.g. [10], [1], [2] and [3]. Although the local structure of a solution of (1) is simple in the way that it is finitely branched about the movable singularities described here, its global structure can be complicated.…”

Polynomial Hamiltonian systems with movable algebraic singularities

“…Lifting the restriction of the Painlevé property, a number of articles have studied classes of second-order differential equations for which all movable singularities are at most algebraic branch points. In [19,20], S. Shimomura considered the equations y ′′ = 2(2k + 1) (2k − 1) 2 y 2k + z, k ∈ N,…”