A. Contreras, D.E. Pelinovsky, and V. Slastikov
Domain walls in the coupled Gross-Pitaevskii equations with the harmonic potential,
Calc Var 61 (2022) 164 (28 pages)
Abstract:
We study the existence and variational characterization of steady states in
a coupled system of Gross-Pitaevskii equations modeling two-component Bose-Einstein
condensates with the magnetic field trapping. The limit with no trapping has been
the subject of recent works where domain walls have been constructed and several
properties, including their orbital stability have been derived. Here we focus on the
full model with the harmonic trapping potential and characterize minimizers according
to the value of the coupling parameter γ. We first establish a rigorous connection
between the two problems in the Thomas-Fermi limit via Γ-convergence. Then, we
identify the ranges of γ for which either the symmetric states (γ < 1) or the uncoupled
states (γ > 1) are minimizers. Domain walls arise as minimizers in a subspace of the
energy space with a certain symmetry for some γ > 1. We study bifurcation of the
domain walls and furthermore give numerical illustrations of our results.
Keywords:
coupled Gross--Pitaevskii equations, domain walls, harmonic potentials,
variational methods, Schrodinger operators, Lyapunov-Schmidt reductions.