Some examples of coupling equations for differential equations of normal form (Algebraic analytic methods in complex partial differential equations)

Abstract

The notion of coupling equations was introduced by H. Tahara [2], for a theory of a class of transformations between some nonlinear partial differential equations in complex domains. In his original coupling theory, solutions to a coupling equation were treated as formal power series of a special form in infinitely many variables. Recently, in collaboration with R. Schäflse and H. Tahara [1], the author studied a functional analytic treatment of coupling equations and showed the unique solvability of their initial value problems. At a glance, in both cases, solutions to coupling equations seem to correspond with linear or nonlinear differential operators of infinite order. But, actually, corresponding transformations are in general not local operators outside the initial surface. In this report, we give some elementary and solvable examples of coupling equations for differential equations of normal form, in order to illustrate such a non-local nature of couplings.