3.4 - Normal forms and Stokes multipliers of nonlinear meromorphic differential equations

Summary

Introduction

Multisummability of formal solutions of meromorphic differential equations was proved by J.-P.Ramis [9], J.Martinet and J.-P.Ramis [8], B.Malgrange and J.-P.Ramis [7], B.L.J.Braaksma [2,3] and W.Balser, B.L.J.Braaksma, J.-P.Ramis and Y.Sibuya [1]. In particular, B.L.J.Braaksma [3] treated nonlinear cases by means of a method based on J.Ecalle's theory of acceleration (cf. J.Ecalle [4] and J.Martinet and J.-P.Ramis [8]). In this paper, we shall outline another proof based on the cohomological definition of multisummability (cf. B.Malgrange and J.-P.Ramis [7] and W.Balser, B.L.J.Braaksma, J.-P.Ramis and Y.Sibuya [1]). The main problem is explained in §2 (cf. Theorem 2.1). In this paper, we shall outline a proof of Theorem 2.1 only, since multisummability of formal power series solutions can be derived from Theorem 2.1 in a manner similar to the proof of Theorem 4.1 based on Lemma 7.1 in paper [1]. In our outline, we shall show mostly the formal part which is the key idea. An analytic justification of the formal part utilizes methods due to M.Hukuhara [5], M.Iwano [6], and J.-P.Ramis and Y.Sibuya [10]. We shall publish another paper (jointly with J.-P.Ramis) in which the entire analysis will be explained in detail.