Cubic-quintic Gross–Pitaevskii equation ($i\hslash {\psi }_t=-{\hslash }^2{\nabla }^2\psi /(2m)+V_0|\psi {|}^2\psi +W_0|\psi {|}^4\psi$)

A local two-body and three-body mean-field energy density $V_0{n}^2/2+W_0{n}^3/3$ generates cubic and quintic terms in the Gross–Pitaevskii equation. Variation with respect to ${\psi }^{\ast }$ gives $(V_{0}n+W_0{n}^2)\psi$, with $n=|\psi {|}^2$. A uniform stationary state's chemical potential is $\mu =V_0{n}_0+W_0{n}_0^2$. Its positive-compressibility condition is $V_0+2W_0{n}_0>0$. If $\mu >0$, scaling amplitude by $\sqrt{n_0}$ and distance by $\hslash /\sqrt{2m\mu }$ gives coefficients $\alpha =V_0{n}_0/\mu$, $\beta =W_0{n}_0^2/\mu$ and $\alpha +\beta =1$. Negative chemical potential requires a different sign in the dimensionless linear term.