2019/02/14 by Luo, Yongming, Stylianou, Athanasios · 1 citation
#35B09 #35Q55 #49J35 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1902.05591
We study the following nonlocal mixed order Gross-Pitaevskii equation i ∂t ψ=-(1)/(2) Δψ+Vext ψ+λ1 |ψ|2 ψ+λ2 (K*|ψ|2) ψ+λ3 |ψ|p-2 ψ, where K is the classical dipole-dipole interaction kernel, λ3>0 and p∈(4,6]; the case p=6 being energy critical. For p=5 the equation is considered currently as the state-of-the-art model for describing the dynamics of dipolar Bose-Einstein condensates (Lee-Huang-Yang corrected dipolar GPE). We prove existence and nonexistence of standing waves in different parameter regimes; for p≠ 6 we prove global well-posedness and small data scattering.