Inverse Problems in the Theory of Singular Perturbations

R. Schäfke

Contemporary Mathematics. Fundamental Directions, Proceedings of the International Conference on Differential and Functional-Differential Equations — Satellite of International Congress of Mathematicians ICM-2002 (Moscow, MAI, 11–17 August, 2002). Part 3, Tome 3 (2003), pp. 63-88

Résumé

First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation $\varepsilon^2y''=(1+\varepsilon^2\psi(x,\varepsilon))y$ with a small parameter $\varepsilon$, where $\psi$ is analytic and even with respect to $\varepsilon$. It is well known that it has two formal solutions of the form $y^\pm(x,\varepsilon)=e^{\pm x/\varepsilon}h^\pm(x,\varepsilon)$, where $h^\pm(x,\varepsilon)$ is a formal series in powers of $\varepsilon$ whose coefficients are functions of $x$. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if $\psi$ satisfies certain conditions, in particular concerning its $x$-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations $\varepsilon^ry'=A(x,\varepsilon)$ in the neighborhood of a nonturning point and for second-order (analytic) equations $\varepsilon y''-2xy'-g(x,\varepsilon)y=0$ exhibiting resonance in the sense of Ackerberg–O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution $\hat y(x,\varepsilon)=\sum y_n(x)\varepsilon^n$ such that the coefficients have no poles at $x=0$.