Painlev\'e Asymptotics of the Focusing Nonlinear Schr\"odinger Equation with a Finite-Genus Algebro-Geometric Background

Abstract

We investigate the Cauchy problem for the focusing nonlinear Schr\"odinger (NLS) equation \begin{equation} iq_t(x,t)+q_{xx}(x,t)+2|q(x,t)|^2q(x,t)=0,\quad x\in\mathbb{R},\quad t\ge0,\nonumber \end{equation} subject to initial data $ q(x,0)$ satisfying the asymptotic boundary conditions \begin{equation}\label{eq:boundary} q(x,0) \sim q^{alg}(x,0) \quad \text{as} \quad x \to \pm\infty,\nonumber \end{equation} where $q^{alg}(x,t)$ denote finite-genus algebro-geometric quasi-periodic solutions of the focusing NLS equation. Employing the Riemann--Hilbert (RH) approach combined with the Deift--Zhou nonlinear steepest descent method, we analyze the long-time asymptotic behavior of solutions to this Cauchy problem. Our analysis distinguishes between two cases based on the genus $n$ of the underlying hyperelliptic Riemann surface: (i) Odd genus backgrounds: When the background solutions $q^{alg}(x,0)$ correspond to hyperelliptic curves of odd genus $n = 2s+1$ $(s \in \mathbb{N}_0)$, we identify distinct asymptotic regions in the $(x,t)$-plane characterized by the variable $\xi = x/t$, within which the leading-order asymptotics is expressed in terms of the second Painlev\'e transcendent. (ii)Even genus backgrounds: When the background solutions $q^{alg}(x,0)$ correspond to hyperelliptic curves of even genus $n = 2s$ $(s \in \mathbb{N})$, the asymptotic behavior in regions selected by $\xi$ is described in terms of parabolic cylinder functions. Specifically, we derive the leading-order asymptotics and establish explicit error bounds for the solution $q(x,t)$ as $t \to +\infty$, uniformly for $x \in \mathbb{R}$.