= Cubic-quintic Gross–Pitaevskii equation
{title2=$i\hbar\psi_t=-\hbar^2\nabla^2\psi/(2m)+V_0|\psi|^2\psi+W_0|\psi|^4\psi$}
A local two-body and three-body mean-field energy density $V_0n^2/2+W_0n^3/3$ generates cubic and quintic terms in the <Gross–Pitaevskii equation>. Variation with respect to $\psi^*$ gives $(V_0n+W_0n^2)\psi$, with $n=|\psi|^2$. A uniform stationary state's <chemical potential> is $\mu=V_0n_0+W_0n_0^2$. Its positive-compressibility condition is $V_0+2W_0n_0>0$. If $\mu>0$, scaling amplitude by $\sqrt{n_0}$ and distance by $\hbar/\sqrt{2m\mu}$ gives coefficients $\alpha=V_0n_0/\mu$, $\beta=W_0n_0^2/\mu$ and $\alpha+\beta=1$. Negative chemical potential requires a different sign in the dimensionless linear term.
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