Nonlinear Stokes phenomena in first or second order differential equations

Abstract

We study singularity formation in nonlinear differential equations of order $m\leqslant 2$, $y^{(m)}=A(x^{-1},y)$. We assume $A$ is analytic at $(0,0)$ and $\partial_y A(0,0)=\lambda\ne 0$ (say, $\lambda=(-1)^m$). If $m=1$ we assume $A(0,\cdot)$ is meromorphic and nonlinear. If $m=2$, we assume $A(0,\cdot)$ is analytic except for isolated singularities, and also that $\int_{s_0}^\infty |\Phi(s)|^{-1/2}d|s|<\infty$ along some path avoiding the zeros and singularities of $\Phi$, where $\Phi(s)=\int_{0}^s A(0,\tau)d\tau$. Let $H_{\alpha}=\{z:|z|>a>0,\arg(z)\in (-\alpha,\alpha)\}$. If the Stokes constant $S^+$ associated to $\RR^+$ is nonzero, we show that all $y$ such that $\lim_{x\to +\infty}y(x)=0$ are singular at $2\pi i$-quasiperiodic arrays of points near $i\RR^+$. The array location determines and is determined by $S^+$. Such settings include the Painlev\'e equations $P_I$ and $P_{II}$. If $S^+=0$, then there is exactly one solution $y_0$ without singularities in $H_{2\pi-\epsilon}$, and $y_0$ is entire iff $y_0=A(z,0)\equiv 0$. The singularities of $y(x)$ mirror the singularities of the Borel transform of its asymptotic expansion, $\mathcal{B}\tilde{y}$, a nonlinear analog of Stokes phenomena. If $m=1$ and $A$ is a nonlinear polynomial with $A(z,0)\not\equiv 0$ a similar conclusion holds even if $A(0,\cdot)$ is linear. This follows from the property that if $f$ is superexponentially small along $\RR^+$ and analytic in $H_{\pi}$, then $f$ is superexponentially unbounded in $H_{\pi}$, a consequence of decay estimates of Laplace transforms.