Nonlinear Stokes Phenomena in First or Second Order Differential Equations Dedicated to Professor Kawai on the Occasion of His 60th Birthday

We study singularity formation in nonlinear differential equations of order m 6 2, y(m) = A(x−1, y). We assume A is analytic at (0, 0) and ∂yA(0, 0) = λ 6= 0 (say, λ = (−1)m). If m = 1 we assume A(0, ·) is meromorphic and nonlinear. If m = 2, we assume A(0, ·) is analytic except for isolated singularities, and also that ∫ ∞ s0 |Φ(s)|−1/2d|s| < ∞ along some path avoiding the zeros and singularities of Φ, where Φ(s) = ∫ s 0 A(0, τ)dτ . Let Hα = {z : |z| > a > 0, arg(z) ∈ (−α, α)}. If the Stokes constant S+ associated to R+ is nonzero, we show that all y such that limx→+∞ y(x) = 0 are singular at 2πi-quasiperiodic arrays of points near iR+. The array location determines and is determined by S+. Such settings include the Painlevé equations PI and PII . If S + = 0, then there is exactly one solution y0 without singularities in H2π−ǫ, and y0 is entire iff y0 = A(z, 0) ≡ 0. The singularities of y(x) mirror the singularities of the Borel transform of its asymptotic expansion, Bỹ, a nonlinear analog of Stokes phenomena. If m = 1 and A is a nonlinear polynomial with A(z, 0) 6≡ 0 a similar conclusion holds even if A(0, ·) is linear. This follows from the property that if f is superexponentially small along R+ and analytic in Hπ, then f is superexponentially unbounded in Hπ , a consequence of decay estimates of Laplace transforms. Compared to [2] this analysis is restricted to first and second order equations but shows that singularities always occur, and their type is calculated in the polynomial case. Connection to integrability and the Painlevé property are discussed.