+2}     = \frac{1}{2(2-s)}   .$$

To obtain a polynomial bound, note that we may take $${N_{\text{HL}}}= C(K, C_1, s, \beta) T^\gamma,$$ where $$\gamma = \max\Bigr\{ \frac{1}{(2s-1)\alpha}, \frac{1}{\beta + \frac12 - s - 2\alpha(1-s)}, \frac1{\alpha + 2\beta -1   },  \frac{1}{2\beta - s    - \alpha(1-s)} \Bigl\}+.$$ Therefore we can take $$N \approx \biggl( T^{\alpha(1-s) \gamma} + T^{\bigl(\frac12-\beta   \bigr) \gamma} \biggr).$$ This allows us to conclude that the nonlinear part of the Zakharov flow grows at most polynomially in the $H^s \times L^2$ norm.

Local Theory Result
===================

The section contains the proof of the required local theory for the difference equation. The statement is as follows.

\[localtheory\] Fix $\delta \ll 1$, $\epsilon_0 \ll 1$, $C_0 > 0$, and input functions $(u,n)$ such that $$\begin{cases}
 \delta^{\frac12-\epsilon_0}\|u\|_{X^{s,\frac12,1}_{\delta}} &\leq C_0 \\
\delta^{\frac12-\epsilon_0}\| n\|_{Y^{0,\frac12,1}_{\delta}} &\leq C_0. 
\end{cases}$$ Consider the difference equation on $[0,\delta]$ given by $$\label{eq:diff_eq}
\begin{cases}
iv_t + \Delta v = {\operatorname{Re}}(n + m)(u + v) - {\operatorname{Re}}(n)u \\
im_t - Dm = D[ |u + v|^2 - |u|^2 ] 
\end{cases}$$ with initial data $$\begin{aligned}
v(x,0)  &= 0 \\
m(x,0) &= m_0 := {\operatorname{W}}(t_0) \Biggl( \sum_{|k| \geq {N_{\text{HL}}}} \frac{h_k}{{\langle}k {\rangle}^\beta} e^{ikx}\Biggr).\end{aligned}$$ We assume that the coefficients $h_k$ satisfy $$\sup_k |h_k| \leq C_1 \quad \text{for some} \quad 