Convergence of formal Dulac series satisfying an algebraic ordinary differential equation

Abstract

A sufficient condition is proposed which ensures that a Dulac series that formally satisfies an algebraic ordinary differential equation (ODE) is convergent. Such formal solutions of algebraic ODEs are quite common: in particular, the Painlevé III, V and VI equations have formal solutions given by Dulac series; they are convergent in view of the sufficient condition presented. Bibliography: 13 titles.

On the Convergence of Formal Exotic Series Solutions of an ODE

“…One can also establish convergence with the use of Theorem 1, rewriting the equation ( 27) in a polynomial form F (x, y, δy, δ 2 y) = 0 and directly checking that the series ∂F/∂y j (x, Φ) begin with (polynomial in ln x) • x 2 . Note that although the formal solution (28) has the form of a Dulac series, the theorem on convergence from the paper [3] formally cannot be applied here, since its assumptions require that the leading term of the partial derivative ∂F/∂y 2 (x, Φ) does not contain logarithm.…”

“…Example 2. In this example we describe a situation in which a formal power-log series solution of an algebraic ODE that satisfies the sufficient condition of convergence from Theorem 1 is produced from a formal Dulac series solution of another algebraic ODE that satisfies the sufficient condition of convergence from [3], via a rational transformation of the unknown.…”

“…Thus, our present goal is to obtain some general condition for the convergence of an exotic formal series solution of the equation (1). In this sense, our work continues a series of articles where similar questions were studied for generalized formal power series [4], [6] and formal Dulac series [7], which were inspired by the original paper of B. Malgrange [9] on the classical formal power series solutions of a non-linear ODE. F (x, Φ) = 0, Φ := (ϕ, δϕ, .…”

“…One may regard them as the generalization of Dulac series (whose coefficients R k are polynomials) and expect that a sufficient condition of the convergence of a formal powerlog series solution is similar to that of a formal Dulac series solution from the paper [3]. Here we use again a universal majorant method but the proof in the present article also contains quite a few new moments compared to [3]: this is in part due to the fact that the coefficients of a formal Dulac series solution of (2) that satisfies the sufficient condition of convergence from [3], starting with some number k are solutions of nonhomogeneous linear ODEs of order n with constant coefficients, whereas the coefficients of power-log series considered here are solutions of nonhomogeneous linear ODEs of order n with rational coefficients. In particular, we will show that the poles of all the coefficients R k of the series (1) that formally satisfies (2), belong to a common finite set and that their orders have a linear growth with respect to k.…”

“…The second step is to prove the convergence of the series (8). Let us write down the equation (7) in the form…”