Overstability and resonance

Abstract

We consider a singularity perturbed nonlinear differential equation εu ' =f(x)u++εP(x,u,ε) which we suppose real analytic for x near some interval [a,b] and small |u|, |ε|. We furthermore suppose that 0 is a turning point, namely that xf(x) is positive if x≠0. We prove that the existence of nicely behaved (as ϵ→0) local (at x=0) or global, real analytic or C ∞ solutions is equivalent to the existence of a formal series solution ∑u n (x)ε 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 εy '' +ϕ(x,ε)y ' +ψ(x,ε)y=0 with ϕ and ψ real analytic for x near some interval [a,b] and small |ε|. We assume that -xϕ(x,0) is positive if x≠0 and that the function ψ 0 :x↦ψ(x,0) has a zero at x=0 of at least the same order as ϕ 0 ↦ϕ(x,0). For this equation, we prove that the existence of local or global, real analytic or C ∞ solutions tending to a nontrivial solution of the reduced equation ϕ(x,0)y ' +ψ(x,0)y=0 is equivalent to the existence of a non trivial formal series solution y ^(x,ε)=∑y n (x)ε 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 y ^ has a prescribed logarithmic derivative y ^ ' (0,ε)/y ^(0,ε) at x=0 which is divergent of Gevrey order 1.