Consider the functional \(S[\phi, \phi^*] = 4\pi \int_{-\infty}^{\infty} dt \int_0^\infty dr\, r^2 \left(\dot{\phi}\dot{\phi}^* - \phi'\phi'^* - U(\phi\phi^*)\right)\), where \(U \in C^1(\mathbb{R})\) and \(\phi\) is a \(C^2(\mathbb{R} \times [0, \infty))\) complex-valued function of \(t\) and \(x\).
Varying with respect to \(\phi\) and \(\phi^*\), obtain the Euler-Lagrange equation and its conjugated. Substituting the ansatz \(\phi(r, t) = f(r) e^{i\omega t}\), rewrite them as an equation on a function \(f\) of a single variable \(r\).
Substitute the ansatz \(\phi(r, t) = f(r) e^{i\omega t}\) into the functional \(S[\phi, \phi^*]\) first, and, varying with respect to \(f\), obtain the Euler-Lagrange equation on \(f\). Compare with the result of the previous point.
The source says \(\phi\) is a function “of \(t\) and \(x\)”, but the functional and the ansatz use the radial variable \(r \in [0, \infty)\), with \(\dot{\phi} = \partial_t \phi\) and \(\phi' = \partial_r \phi\); the intended variables are \(t\) and \(r\).