Consistent derivation of the fractional Gross–Pitaevskii equation

Derive a consistent mean-field fractional Gross–Pitaevskii equation for Bose–Einstein condensates whose single-particle dynamics follow the fractional Schrödinger equation with Lévy index α, yielding an evolution equation of the form i ∂Ψ/∂t = (1/2)(−∇²)^{α/2} Ψ + V(x,y) Ψ + σ|Ψ|²Ψ, where σ = +1 (repulsive interactions) or −1 (attractive interactions).

Background

The paper discusses fractional quantum mechanics, where particle dynamics are governed by the fractional Schrödinger equation with a Riesz fractional Laplacian determined by the Lévy index α. In the mean-field description of Bose–Einstein condensates, interactions are typically captured by the cubic Gross–Pitaevskii equation.

Extending this framework to fractional dynamics motivates a fractional Gross–Pitaevskii equation (GPE). However, the authors explicitly state that a consistent derivation of such a mean-field model remains unresolved. They nevertheless present the expected scaled form of the equation, which augments the fractional kinetic term with a cubic nonlinearity representing inter-particle interactions, and indicate the roles of repulsive and attractive interactions via the sign parameter σ.