Electron. J. Diff. Eqns., Vol. 2002(2002), No. 51, pp. 1-27.
On combined asymptotic expansions in singular perturbations
Eric Benoit, Abdallah El Hamidi, & Augustin Fruchard
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 {\it 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 Schr\"{o}dinger
equation in the double well symmetric case. The structured nature
of our approach allows us to give effective symbolic algorithms.
Submitted March 10, 2002. Published June 3, 2002.
Math Subject Classifications: 34E05, 34E15, 34E18, 34E20.
Key Words: Singular perturbation, combined asymptotic expansion,
turning point, canard solution.
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Eric Benoit
Laboratoire de Mathematiques
Universite de La Rochelle
Pole Sciences et Technologie
Avenue Michel Crepeau
17042 La Rochelle cedex 1, France
email: ebenoit@univ-lr.fr |
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Abdallah El Hamidi
Laboratoire de Mathematiques
Universite de La Rochelle
Pole Sciences et Technologie
Avenue Michel Crepeau
17042 La Rochelle cedex 1, France
email: aelhamid@univ-lr.fr |
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Augustin Fruchard
Laboratoire de Mathematiques
Universite de La Rochelle
Pole Sciences et Technologie
Avenue Michel Crepeau
17042 La Rochelle cedex 1, France
email: afruchar@univ-lr.fr |
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