Collective synchronization of the multi-component Gross–Pitaevskii–Lohe system

Abstract

Abstract In this paper, we propose a multi-component Gross–Pitaevskii–Lohe (GPL for brevity) system in which quantum units interact with each other such that collective behaviors can emerge asymptotically. We introduce several sufficient frameworks leading to complete and practical synchronizations in terms of system parameters and initial data. For the modeling of interaction matrices we classify them into three types (fully identical, weakly identical and heterogeneous) and present emergent behaviors correspond to each interaction matrix. More precisely, for the fully identical case in which all components are same, we expect the emergence of the complete synchronization with exponential convergence rate. On the other hand for the remaining two interaction matrices, we can only show that the practical synchronization occurs under well-prepared initial frameworks. For instance, we assume that a coupling strength is sufficiently large and perturbation of an interaction matrix is sufficiently small. Regarding the practical synchronization estimates, due to the possible blow-up of a solution at infinity, we a priori assume that the L 4 -norm of a solution is bounded on any finite time interval. In our analytical estimates, two-point correlation function approach will play a key role to derive synchronization estimates. We also provide several numerical simulations using time splitting Crank–Nicolson spectral method and compare them with our analytical results.