Gross–Pitaevskii equation

Remarks

For $V = 0$ it is, after scaling, the cubic nonlinear Schrödinger equation. A standing wave $\psi(t, x) = e^{-i\mu t/\hbar} \phi(x)$ solves it exactly when $\phi$ solves the stationary equation $\mu \phi = -\tfrac{\hbar^2}{2m} \Delta \phi + V \phi + g \abs{\phi}^2 \phi$, and $\mu$ is the chemical potential.

Example: for $V = 0$ and $n > 0$, the constant state $\phi = \sqrt{n}$ solves the stationary equation with $\mu = g n$, since $\Delta \phi = 0$ and $g \abs{\phi}^2 \phi = g n \phi$.

Non-example: if $V$ is not constant, then no nonzero constant $\phi$ solves the stationary equation, because $\mu \phi = (V(x) + g \abs{\phi}^2) \phi$ would force $V$ to be constant.

The ground state minimises the Gross–Pitaevskii energy $\int \big( \tfrac{\hbar^2}{2m} \abs{\nabla \phi}^2 + V \abs{\phi}^2 + \tfrac{g}{2} \abs{\phi}^4 \big) dx$ under the constraint $\int \abs{\phi}^2 \, dx = N$, the number of particles.