Classification of resonant equations

Augustin Fruchard (1) , Reinhard Schäfke (2)

1 LMIA - Laboratoire de Mathématiques Informatique et Applications [UHA]
2 IRMA - Institut de Recherche Mathématique Avancée

We consider singularly perturbed linear ordinary differential equations of the second order with coefficients analytic near some point, say 0. We assume that the coefficients are real valued on the real axis, i.e. that there is a turning point at the origin. Such equations are called resonant in the sense of Ackerberg-O'Malley, if there is a solution, analytic in some neighborhood of 0, which tends to a non-zero limit as the parameter tends to 0. The article presents a classification of such resonant equations with respect to linear transformations having analytic coefficients. Besides a formal invariant (considered fixed below), we associate three formal series of Gevrey order 1 to any resonant equation which are invariant under analytic transformations. It is shown that this correspondence between equivalence classes of resonant equations and triples of Gevrey 1 series is essentially bijective, and that each equivalence class contains an equation of a particular form.

Mots clés: Ackerberg-O'Malley resonance; Gevrey asymptotic expansions; singular perturbation

Domaines: Analyse classique [math.CA]

Augustin Fruchard, Reinhard Schäfke. Classification of resonant equations. Journal of Differential Equations, 2004, 207 (2), pp.360-391. ⟨10.1016/j.jde.2004.06.016⟩. ⟨hal-00144867⟩