Unique solvability of coupling equations in holomorphic functions (Several aspects of microlocal analysis)

Abstract

The theory of coupling equations was introduced by the third author [4], as a theory of a class of transformations between some nonlinear partial differential equations in complex domains. There, he constructed, to the initial value problem of a coupling equation, a formal power series solution of a special form in infinitely many variables, satisfying suitable estimates. It would be desirable, from several aspects, to study the coupling equations and their solvability as functional equations for holomorphic functions . In this report, we consider coupling equations for partial differential equations of normal form in the t variable. After preparing and recalling some notions of holomorphy on infinite dimensional spaces, we announce our recent result on the unique solvability of the initial value problem of a coupling equation, using the contraction mapping principle.