Existence and dynamical behaviour of vectorial standing waves with prescribed mass for Hartree-Fock type systems
arxiv(2024)
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.
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