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

TL;DR

This paper analyzes the long-time behavior of focusing NLS solutions with finite-genus backgrounds, revealing Painlevé transcendents or parabolic cylinder functions as leading asymptotics depending on the genus parity.

Contribution

It provides a detailed asymptotic description of focusing NLS with finite-genus backgrounds using Riemann-Hilbert methods, distinguishing odd and even genus cases.

Findings

Asymptotics for odd genus involve Painlevé II transcendents.

Asymptotics for even genus involve parabolic cylinder functions.

Explicit error bounds are established for large time behavior.

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 $x \in \mathbb{R}$.