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

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 $|ψ^{\eps}|^{2}$ converges toward the chemical potential $ρ_{0}(x)$ 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.

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