Suppose that $u: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$ is a classical solution to the nonlinear Schrödinger equation: \begin{align*} iu_t+\frac{1}{2}\Delta u = -u|u|^2. \tag{1} \end{align*} Let $a: \mathbb{R}^N \to \mathbb{R}$ be a smooth function of at most polynomial growth. By multiplying (1) by $\bar{u}$, taking the imaginary part and integrating by parts, we can obtain the following identity: \begin{align*} \partial_t \int_{\mathbb{R}^n}a(x) |u|^2(x,t)dx = \int_{\mathbb{R}^n}\partial_{x_j}a(x) Im(\bar{u}\partial_{x_j}u)(x,t) dx. \tag{2} \end{align*} where we are using the summation convention on the indices. Within chapter 3 of these notes on nonlinear dispersive equations, it seems through further properties of the equation and some integration by parts we can obtain a Virial identity: \begin{align*} \partial^2_t \int_{\mathbb{R}^n}a(x) |u|^2(x,t)dx = \int_{\mathbb{R}^n}\partial_{x_j}\partial_{x_k}a(x)Re(\partial_{x_j}u\overline{\partial_k u})(x,t) &- \frac{1}{2} |u(x,t)|^4\Delta a(x) \\&- \frac{1}{4}|u(x,t)|^2\Delta^2a(x)dx. \tag{3} \end{align*} I am unable to derive (3). The linked notes state (and leave the proof an exercise) the following identity which they say can be derived from (1): \begin{align*} \partial_t Im(\bar{u}\partial_{x_j}u) + \partial_{x_k}Re(\partial_{x_j}u\overline{\partial_k u})-\frac{1}{4}\delta_{jk}\Delta(|u|^2) - \frac{1}{4} \delta_{jk}|u|^4= 0, \tag{4} \end{align*} and I am able to deduce (3) from (4) through integration by parts. However, I cannot prove (4). My question is then: how do we deduce (4) from (1)? Alternatively, some other computational or arithmetic way of deducing (3) would be desirable.
There are several places (such as Tao's notes) where the Virial identity is mentioned, however these state that (3) follows from manipulation of (1) and integration parts, and my problem is being able to realise the details of this computation .