A dense focusing Ablowitz-Ladik soliton gas and its asymptotics

TL;DR

This paper introduces a soliton gas solution for the focusing Ablowitz-Ladik system, analyzing its asymptotic behavior through a Fredholm determinant and Riemann-Hilbert approach, revealing detailed large-time and large-space dynamics.

Contribution

It presents the first soliton gas solution for the focusing Ablowitz-Ladik system, derived as a large N limit with a novel spectral and asymptotic analysis.

Findings

Fredholm determinant representation of the gas solution

Large-space asymptotics at t=0 established

Large-time asymptotics characterized

Abstract

In this paper, we propose a soliton gas solution for the focusing Ablowitz-Ladik system. This solution is defined as the large N limit of the N-soliton solution, and arises from a continuous spectrum of poles that accumulate within two disjoint intervals on the imaginary axis. We show that this gas solution admits a Fredholm determinant representation. By further exploring its Riemann-Hilbert characterization, we are able to establish the large-space asymptotics at t = 0 and large-time asymptotics of the gas solution.