On combined asymptotic expansions in singular perturbations

Eric Benoît (1) , Abdallah El Hamidi (1) , Augustin Fruchard (1)

1 LMCA - Laboratoire de Mathématiques Calcul Asymptotique

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 canard 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.

Domaines: Equations aux dérivées partielles [math.AP]

Eric Benoît, Abdallah El Hamidi, Augustin Fruchard. On combined asymptotic expansions in singular perturbations. Electronic Journal of Differential Equations, 2002, 2002 (51), pp.1 - 27. ⟨hal-02507103⟩