Existence and dynamical behaviour of vectorial standing waves with prescribed mass for Hartree-Fock type systems

TL;DR

This paper studies the existence, symmetry, and stability of vectorial standing waves with prescribed mass in Hartree-Fock systems, revealing conditions for their behavior and connection to quantum mechanical models.

Contribution

It provides new results on the existence, symmetry, and stability of vectorial standing waves in Hartree-Fock systems with prescribed mass, including conditions for blow-up and stability.

Findings

Existence and nonexistence of vectorial energy ground states.

Symmetry properties of ground states.

Conditions for global well-posedness and blow-up.

Abstract

In this paper, we investigate vectorial standing waves with prescribed mass for the Hartree-Fock type system (HF system) with the double coupled feature. Such system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics. By exploring the interaction of the double coupled terms, we prove the exis?tence/nonexistence and symmetry of vectorial energy ground states for the corresponding stationary problem. Furthermore, we obtain the relation between vectorial energy ground states and vectorial action ground states in some cases. Finally, we establish conditions for global well-posedness and finite time blow-up to HF system with the initial data, and prove orbital stability/strong instability of standing waves.