 Solution

Since each species number $N_i=\int |{\psi }_i{|}^2{d}^{3}x$ is separately conserved, minimize $E-{\mu }_1{N}_1-{\mu }_2{N}_2$ with two independent Lagrange multipliers. Variation with respect to each complex-conjugate field, with boundary terms removed, givesThese are the two stationary Gross–Pitaevskii equations; the time-dependent equilibrium fields are $e^{-i{\mu }_{i}t/\hslash }{\psi }_i(\mathbf{x})$. The cross term contributes $U_{12}|{\psi }_j{|}^2{\psi }_i$ to each equation, with no factor two, while differentiating the self-interaction term cancels its factor $1/2$. Separate number conservation does not require the two chemical potentials to be equal.