Consider a complex-valued solution $A= A(z, x, \tau)$, $(z,x,\tau) \in \mathbb{R} \times [0, \infty)$ to a nonlinear Schrödinger-type equation
$$-i\partial_z A + \Delta_{x,\tau} A - |A|^2A, \tag{*} \label{NLS}$$
where $\Delta_{x, \tau} = \partial^2_x + \partial^2_\tau$. Versions of this equation are used to model, for instance, the intensity of a pulsed laser beam ($z$ denotes the propagation axis of the laser beam, $x$ the "profile" axis, and $\tau$ time). The initial data lives on $\{z = 0\}$, which we assume belongs to the Schwartz class in $(x, \tau)$.
I aim to study two quantities associated to the solution of \eqref{NLS}. The first is the "energy" $$E = E(z) = \int \int |\nabla_{x,\tau} A|^2 - \frac{1}{2} |A|^4 d\tau dx,$$ and the second the "moment" $$P = P(z) = \int \int (x^2 + \tau^2) |A|^2 d\tau dx.$$
In particular, I want to show $\partial_z E = 0$ and $\partial^2_z P = k E$ for some constant $k > 0$.
The significance these identities is that if $E(0) < 0$ then $P(z)$, as it obeys as a parabolic trajectory, reaches $P = 0$ in finite $z$. Thus at this $z$-value the solution $A$ must be wholly supported at the origin. This phenomenon is known as "wave collapse".
I believe the above identities are "well-known" but I cannot find a straightforward argument anywhere in the literature, so I am trying to reproduce from scratch. I have also heard that the choice of "sign" in \eqref{NLS} (i.e., $-|A|^2A$ versus $+|A|^2A$) is extremely important to the behavior of solutions to the nonlinear Schrödinger equation. But I'm completely ignorant of such things so please correct if there is a grave error above.
I will proceed formally, assuming always that $z \mapsto A(z, \cdot, \cdot )$ is smooth with values in the Schwartz class in $x$ and $\tau$. That way, I can freely reorder any partial derivatives and ignore boundary terms when I integrate by parts.
Here is my attempt. Split the solution to \eqref{NLS} into its real and imaginary parts: \begin{equation*} A = u + iv, \end{equation*} which implies \begin{equation*} (u^2 + v^2) u = v_z + u_{xx} + u_{\tau \tau}, \qquad (u^2 + v^2) v = -u_z + v_{xx} + v_{\tau \tau}. \end{equation*} Thus \begin{equation*} E = \int \int u^2_x + v^2_x + u^2_\tau + v^2_\tau - \frac{1}{2} (u^2 + v^2)^2 dx d\tau, \end{equation*} and \begin{equation*} \begin{split} \frac{dE}{dz} &= \int \int 2 u_{xz} u_x + 2 v_{xz} v_x + 2 u_{tz} u_t + 2 v_{tz} v_t- (u^2 + v^2)(2uu_z + 2vv_z) dx d\tau \\ &= 2 \int \int u_{xz} u_x + v_{xz} v_x + u_{\tau z} u_\tau + v_{\tau z} v_\tau- (u_{xx} + u_{\tau \tau} + v_z)u_z - ( v_{xx} + v_{\tau \tau}-u_z)v_z dx d\tau \\ &= \int \int (u_x u_z)_x + (u_\tau u_z)_\tau + (v_x v_z)_x + (v_\tau v_z)_\tau dxd\tau \\ & = 0. \end{split} \end{equation*}
When I compute $\partial^2_z P$, I reorder some derivatives, substitute for $u_z$ and $v_z$, and integrate by parts. This yields $$\partial^2_z P = \int \int (-4xv_x - 4tv_t)(\Delta_{x, \tau} v - N[v]) + (8u + 4xu_x + 4tu_t)(-\Delta_{x, \tau}u + N[u]) dx d\tau, $$
where I condensed notation by putting $N[v] = (u^2 + v^2)v$ and $N[v] = (u^2 + v^2)u$. Should I keep going with this calculation, or is there an error I have made, or is this calculation doomed from the outset because of a sign issue in \eqref{NLS}? I just don't have intuition here. Any hints or solutions are greatly appreciated.