We consider the equation (NLS)
\begin{eqnarray}\label{gnls}
i \epsilon\partial_t u^{\epsilon} + \frac{\epsilon^2}{2}\Delta_{\eta}u^{\epsilon} = \epsilon |u^{\epsilon}|^{2}u^{\epsilon}, \quad x \in \mathbb R^d
\end{eqnarray}
$d\geq 1$ The initial data is supposed to be given by a
superposition of highly oscillatory plane waves, i.e.
\begin{eqnarray}\label{icscm}
u_0^{\epsilon}(x) = \sum_{j\in J_0} \alpha_j(x) e^{ik_j\cdot x/ \epsilon},
\end{eqnarray}
where for some index set $J_0 \subset \mathbb N$ we are given initial wave vectors

$\Phi_0= \{ k_j \in \mathbb R^d: j\in J_0\}$ with corresponding smooth, rapidly decaying amplitudes $\alpha_j \in \mathcal{S}(\mathbb R^d).$
We seek an approximation of the exact solution $u^{\epsilon}$ in the following form
\begin{eqnarray}\label{as}
u^{\epsilon}(t,x)\sim_{\epsilon \to 0} u^{\epsilon}_{app} (t,x) = \sum_{j\in J} a_{j} (t,x) e^{i \phi_j(t,x)/\epsilon} \ \quad (*)
\end{eqnarray}

I'm trying to understand the following paragraph, p. 8-9:

Formally, we plugging the approximation (*) into (NLS), and comparing equal powers of $\epsilon$, we find that the leading order term is of order $ \mathcal{O}(\epsilon^0).$ It can be made identically zero, if for all $j\in J$ \begin{eqnarray} \partial_t\phi_j + \frac{1}{2} |\Delta \phi_j|^2=0. \end{eqnarray}

Questions are: (A) What is the leading order term of equation that is formed after plugging the approximation (*) into NLS ? (B) Why it can be made zero if if for all $j\in J$ \begin{eqnarray} \partial_t\phi_j + \frac{1}{2} |\nabla \phi_j|^2=0. \end{eqnarray}