Bifurcation analysis of stationary solutions of two-dimensional coupled Gross-Pitaevskii equations using deflated continuation

TL;DR

This paper applies the deflated continuation method to analyze the complex solution landscape of two coupled two-dimensional Gross-Pitaevskii equations, revealing numerous bifurcations and stability regimes in ultracold atom models.

Contribution

It extends the deflated continuation technique to coupled Gross-Pitaevskii equations, uncovering a rich variety of solutions and bifurcations in a high-dimensional nonlinear system.

Findings

Identified multiple solution branches including known and novel ones.

Constructed bifurcation diagrams and analyzed spectral stability.

Discovered numerous bifurcations, especially pitchfork types.

Abstract

Recently, a novel bifurcation technique known as the deflated continuation method (DCM) was applied to the single-component nonlinear Schr\"odinger (NLS) equation with a parabolic trap in two spatial dimensions. The bifurcation analysis carried out by a subset of the present authors shed light on the configuration space of solutions of this fundamental problem in the physics of ultracold atoms. In the present work, we take this a step further by applying the DCM to two coupled NLS equations in order to elucidate the considerably more complex landscape of solutions of this system. Upon identifying branches of solutions, we construct the relevant bifurcation diagrams and perform spectral stability analysis to identify parametric regimes of stability and instability and to understand the mechanisms by which these branches emerge. The method reveals a remarkable wealth of solutions: these do not only include some of the well-known ones including, e.g., from the Cartesian or polar small amplitude limits of the underlying linear problem but also a significant number of branches that arise through (typically pitchfork) bifurcations. In addition to presenting a ``cartography'' of the landscape of solutions, we comment on the challenging task of identifying {\it all} solutions of such a high-dimensional, nonlinear problem.