4. Combined Asymptotic Expansions

A structured and synthetic presentation of Vasil'eva's [7] combined expansions is proposed. These expansions simultaneously take into account the limit layer and the slow motion of solutions of a singularly perturbed differential equation. An asymptotic formula is established which gives the distance between two exponentially close solutions. An “input-output” relation around a canard solution is carried out in the case of a turning point. Finally, the distance between two canard values of differential equations with a parameter is given. We illustrate this study on the Liouville equation and the splitting of energy levels in the one dimensional steady Schrödinger equation in the symmetric double well case. The structured nature of our approach allows us to give effective symbolic algorithms. This paper is a short version of [1]. 4.1 Introduction Combined asymptotic expansions are studied in the book [7]. They are used to study singularly perturbed ordinary differential equations. Due to their complexity, turning points are avoided in the basic literature. In this paper, we use combined asymptotic expansions to study the global behaviour of solutions around a turning point. The domain of a combined asymptotic expansion is divided into two parts: the boundary, or inner, layer with its inner expansion, and the outer expansion valid outside the layer. In some of the literature, the two domains are connected using matching techniques, combined asymptotic expansions give approximation formulae valid in large domains containing the boundary layers and the turning point.