Anelastic Approximation of the Gross-Pitaevskii equation for General Initial Data

TL;DR

This paper rigorously analyzes the anelastic approximation of the Gross-Pitaevskii equation with variable chemical potential, showing convergence of density and velocity to the anelastic system as Planck's constant approaches zero.

Contribution

It provides a rigorous mathematical proof of the anelastic approximation for the Gross-Pitaevskii equation with general initial data and variable chemical potential.

Findings

Density converges to the chemical potential as e0 0.

Velocity field converges to the anelastic system.

Resonant effects are overcome by oscillation-cancellation.

Abstract

We perform a rigorous analysis of the anelastic approximation for the Gross-Pitaevskii equation with $x$-dependent chemical potential. For general initial data and periodic boundary condition, we show that as $\eps\to 0$, equivalently the Planck constant tends to zero, the density $|\psi^{\eps}|^{2}$ converges toward the chemical potential $\rho_{0}(x)$ and the velocity field converges to the anelastic system. When the chemical potential is a constant, the anelastic system will reduce to the incompressible Euler equations. The resonant effects the singular limit process and it can be overcome because of oscillation-cancelation.