Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 84 / 1 / i / Solution

Treat $\psi$ and its complex conjugate as independent variables in the functional derivative. Integrating the gradient variation by parts, with vanishing boundary terms, givesThe factors follow from ${\partial }_{{\psi }^{\ast }}|\psi {|}^4=2|\psi {|}^2\psi$ and ${\partial }_{{\psi }^{\ast }}|\psi {|}^6=3|\psi {|}^4\psi$. Thus the cubic-quintic Gross–Pitaevskii equation is