On combined asymptotic expansions in singular perturbation

Eric Benoît, Abdallah El Hamidi and Augustin Fruchard

Université de la Rochelle

Avenue Michel Crépeau

17042 La Rochelle Cedex 1

France

A structured and synthetic presentation of Vasil'eva's combined expansions is proposed. These expansions take into account at once 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 turning point. At last, the distance between two canard values of differential equations with parameter is given. We illustrate this study on Liouville equation and 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.