Painlev\'{e} XXXIV asymptotics for the defocusing nonlinear Schr\"odinger equation with a finite-genus algebro-geometric background

TL;DR

This paper analyzes the long-time behavior of solutions to the defocusing nonlinear Schrödinger equation with a finite-genus background, revealing region-specific decay rates and connections to Painlevé XXXIV transcendents.

Contribution

It provides the first detailed asymptotic description in transition regions involving Painlevé XXXIV functions for this class of solutions.

Findings

Leading order matches background solution with parameter shift

Subleading decay rates vary across regions

Transition regions involve Painlevé XXXIV transcendents

Abstract

In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order $\mathcal{O}(t^{-1/3})$ and the coefficients involve an integral of the Painlev\'e XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann-Hilbert problems.