Numerical Identification of Stationary States and Their Stability in a Model of Quantum Droplets

TL;DR

This paper develops numerical methods to identify and analyze stationary states and their stability in quantum droplet models governed by a modified nonlinear Schrödinger equation, revealing complex bifurcation phenomena.

Contribution

It introduces homotopy grid and dimension-by-dimension homotopy methods to detect novel stationary states and bifurcations in quantum droplet models with quantum fluctuation effects.

Findings

Detected many previously unreported stationary states.

Observed unusual bifurcation events like nonstandard pitchforks.

Identified pathways connecting vortex and dark soliton states.

Abstract

In this work, we are motivated by a recent variant of the nonlinear Schrodinger (NLS) equation describing cold, dilute atomic condensates with quantum fluctuation effects. Our goal is to develop robust numerical methods capable of uncovering diverse stationary solutions in such NLS models. Specifically, and in line with recent theoretical and experimental interest, we focus on ultracold quantum droplets in Bose mixtures influenced by the Lee Huang Yang quantum fluctuation correction and study these systems in one and two dimensional settings. To this end, we deploy several numerical techniques. The homotopy grid method allows systematic refinement from coarse to fine spatial discretizations in one dimension, while the dimension by dimension homotopy approach extends one-dimensional solutions to two-dimensional domains. These methods effectively detect broad families of stationary states, many of which have not been previously reported, to the best of our knowledge. Furthermore, they enable the monitoring of solution continuation and bifurcation phenomena. During our investigation, we encounter unusual bifurcation events, including nonstandard pitchforks and saddle-center bifurcations, which exhibit novel stability transitions. For example, we identify continuous pathways connecting vortex and dark soliton stripe branches, absent in the standard cubic defocusing model. Overall, the presence of competing mean-field and quantum fluctuation interactions leads to a richer bifurcation structure than in traditional cubic NLS systems. These findings suggest that similar complex bifurcation and stability phenomena may appear in other settings, including higher-dimensional systems or models with competing nonlinearities such as cubic-quintic interactions, highlighting the importance of further theoretical and numerical exploration.