Transition asymptotics for the real solutions of the sinh-Gordon Painlevé III equation

Abstract

We consider solutions of the sinh-Gordon Painlevé III equation \[ u_{xx} + \frac{1}{x} u_x = \sinh u \] that are real on $(0,\infty )$. They are parametrized by the monodromy parameter $p\in \overline{\mathbb{C}}$, $|p|>1$, and an additional real parameter $s^{\mathbb{R}}$ when $p=\infty$. Our previous joint work with A. Its described the asymptotic behavior of these solutions as $x\to \infty$. Here, we describe the transition as $x,p\to \infty$, $2\Im (p)=-s^{\mathbb{R}}$, between singular solutions ($|p|<\infty$) and smooth solutions ($p=\infty$). In short, if we parametrize $|p{|}^2=1+e^{2\varkappa x}$, then the smooth exponential asymptotics of the solutions extends to the region $\varkappa >1$, with a change of the leading order term at $\varkappa =2$; at $\varkappa =1$ the exponential behavior transitions into an elliptic asymptotics, which holds for all $0<\varkappa <1$; as $\varkappa$ decays to zero, elliptic asymptotics degenerates into trigonometric one, which holds for all $p$ fixed.