Overstability and resonance

Fruchard, Augustin; Schäfke, Reinhard

Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 227-264.

On considère l’équation différentielle non linéaire singulièrement perturbée $\epsilon u^{\text{'}}=f\left(x\right)u+\epsilon P(x,u,\epsilon )$ qu’on suppose réelle et analytique pour $x$ proche de $[a,b]$ et $\left|u\right|$, $\left|\epsilon \right|$ asez petits. On suppose que 0 est un point tournant, c’est-à-dire $xf\left(x\right)>0$ si $x\ne 0$. On démontre que l’existence de solutions locales (en $x=0$) ou globales, analytiques réelles ou $C^{\infty }$ bornées quand $\epsilon \to 0$ est équivalente à l’existence d’une solution série formelle $\sum u_n\left(x\right){\epsilon }^n$ avec $u_n$ analytiques en $x=0$. L’outil principal de la démonstration est un nouveau “principe de prolongement analytique” pour de telles solutions dites surstables. On applique ce résultat à l’équation d’ordre deux $\epsilon y^{\text{'}\text{'}}+\varphi (x,\epsilon )y^{\text{'}}+\psi (x,\epsilon )y=0$ où $\varphi$ et $\psi$ sont analytiques réelles pour $x$ proche de $[a,b]$ et $\left|\epsilon \right|$ assez petit. On suppose que $-x\varphi (x,0)>0$ si $x\ne 0$ et que la fonction ${\psi }_0:x\mapsto \psi (x,0)$ a un zéro en $x=0$ d’ordre au moins égal à celui de ${\varphi }_0:x\mapsto \varphi (x,0)$. On montre que l’existence de solutions locales ou globales, analytiques réelles ou $C^{\infty }$, tendant vers une solution non triviale de l’équation réduite $\varphi (x,0)y^{\text{'}}+\psi (x,0)y=0$ est équivalente à l’existence d’une solution série formelle non triviale $\widehat{y}(x,\epsilon )=\sum y_n\left(x\right){\epsilon }^n$ avec $y_n$ analytiques en $x=0$. Ceci améliore et généralise un résultat de C.H. Lin concernant le phénomène de “résonance au sens d’Ackerberg-O’Malley”. Dans le dernier paragraphe, on construit des exemples d’ordre deux qui présentent une résonance et tels que la solution formelle $\widehat{y}$ ait une dérivée logarithmique prescrite $\widehat{y}(0,\epsilon )/\widehat{y}(0,\epsilon )$ en $x=0$, divergente d’ordre Gevrey 1.

We consider a singularity perturbed nonlinear differential equation $\epsilon u^{\text{'}}=f\left(x\right)u++\epsilon P(x,u,\epsilon )$ which we suppose real analytic for $x$ near some interval $[a,b]$ and small $\left|u\right|$, $\left|\epsilon \right|$. We furthermore suppose that 0 is a turning point, namely that $xf\left(x\right)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $ϵ\to 0$) local (at $x=0$) or global, real analytic or $C^{\infty }$ solutions is equivalent to the existence of a formal series solution $\sum u_n\left(x\right){\epsilon }^n$ with $u_n$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions. We apply this result to the second order linear differential equation $\epsilon y^{\text{'}\text{'}}+\varphi (x,\epsilon )y^{\text{'}}+\psi (x,\epsilon )y=0$ with $\varphi$ and $\psi$ real analytic for $x$ near some interval $[a,b]$ and small $\left|\epsilon \right|$. We assume that $-x\varphi (x,0)$ is positive if $x\ne 0$ and that the function ${\psi }_0:x\mapsto \psi (x,0)$ has a zero at $x=0$ of at least the same order as ${\varphi }_0\mapsto \varphi (x,0)$. For this equation, we prove that the existence of local or global, real analytic or $C^{\infty }$ solutions tending to a nontrivial solution of the reduced equation $\varphi (x,0)y^{\text{'}}+\psi (x,0)y=0$ is equivalent to the existence of a non trivial formal series solution $\widehat{y}(x,\epsilon )=\sum y_n\left(x\right){\epsilon }^n$ with $y_n$ analytic at $x=0$. This improves and generalizes a result of C.H. Lin on this so-called " Ackerberg-O’Malley resonance" phenomenon. In the proof, the problem is reduced to the preceding problem for the corresponding Riccati equation In the final section, we construct examples of such second order equations exhibiting resonance such that the formal solution $\widehat{y}$ has a prescribed logarithmic derivative ${\widehat{y}}^{\text{'}}(0,\epsilon )/\widehat{y}(0,\epsilon )$ at $x=0$ which is divergent of Gevrey order 1.

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