On Combined Asymptotic Expansions in Singular Perturbations

Abstract

A structured and synthetic presentation of Vasil'eva's combined expansions is proposed. These expansions 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 <i>canard</i> solution is carried out in the case of turning points. We also study the distance between two canard values of differential equations with given parameter. We apply our study to the Liouville equation and to the splitting of energy levels in the one-dimensional steady Schrödinger equation in the double well symmetric case. The structured nature of our approach allows us to give effective symbolic algorithms.