Analysis of multistability in discrete quantum droplets and bubbles

TL;DR

This paper explores the stability and multistability of localized states in discrete quantum droplets and bubbles modeled by the DNLS equation, revealing the mechanisms of homoclinic snaking and the effects of coupling strength.

Contribution

It introduces analytical and numerical methods to analyze the pinning region and multistability in discrete quantum systems with quadratic and cubic nonlinearities.

Findings

Pinning region width depends algebraically on coupling in weak limit.

Pinning region width depends exponentially on coupling in strong limit.

Analytical results show excellent agreement with numerical simulations.

Abstract

This study investigates the existence and stability of localized states in the discrete nonlinear Schr\"odinger (DNLS) equation with quadratic and cubic nonlinearities, describing the so-called quantum droplets and bubbles. Those states exist within an interval known as the pinning region, as we vary a control parameter. Within the interval, multistable states are connected through multiple hysteresis, called homoclinic snaking. In particular, we explore its mechanism and consider two limiting cases of coupling strength: weak (anti-continuum) and strong (continuum) limits. We employ an asymptotic and a variational method for the weak and strong coupling limits, respectively, to capture the pinning region's width. The width exhibits an algebraic and an exponentially small dependence on the coupling constant for the weak and strong coupling, respectively. This finding is supported by both analytical and numerical results, which show excellent agreement. We also consider the modulational instability of spatially uniform solutions. Our work sheds light on the intricate interplay between multistability and homoclinic snaking in discrete quantum systems, paving the way for further exploration of complex nonlinear phenomena in this context.