Nonlinear Stokes phenomena in first or second order differential equations

Abstract

We study singularity formation in nonlinear differential equations of order $m⩽2$, $y^{(m)}=A(x^{-1},y)$. 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 Painlevé 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.