wave part notation for simplicity.

Acknowledgments {#acknowledgments .unnumbered}
===============

The author is grateful to Professor G. Staffilani for many helpful discussions. This work was supported by NSF MSPRF \#1704865.

Proof of Theorem \[mainThm\]
============================

The idea of the proof is to combine the high-low method of Bourgain with an I-method argument. In the following, ${N_{\text{HL}}}\gg 1$ is the frequency cut-off associated with the high-low method part of the argument, and ${N_{\text{I}}}\gg 1$ is that associated with the I-method portion of the argument. These parameters will be fixed later.

Begin by splitting the initial data $n_0^\pm$ into high and low parts: $$\begin{aligned}
n^\pm_0(x) =  P_{\leq {N_{\text{HL}}}} n_0^\pm + P_{n > {N_{\text{HL}}}}n_0^\pm =: n_0^{\pm \text{L}}(x) + n_0^{\pm \text{H}}(x).\end{aligned}$$ Here $P_{\leq {N_{\text{HL}}}}$ denotes the spatial frequency projection onto frequencies of magnitude at most ${N_{\text{HL}}}$, i.e. $${\widehat}{P_{\leq {N_{\text{HL}}}} f} := \chi_{ \{|k| \leq {N_{\text{HL}}}\}} {\widehat}{f} \quad \text{and} \quad  P_{> {N_{\text{HL}}}} := \operatorname{Id} - P_{\leq {N_{\text{HL}}}}.$$

Let $(u^1,n^{\pm1})$ be the solution to with initial data $(u_0, n_0^{\pm \text{L}})$. To obtain this solution, we use the local theory adapted to the I-method from [@Kish2 Prop. 4.5]. This says that we can obtain a solution $(u^1,n^{\pm1}) \in X^{s,\frac12,1}_\delta \times Y^{0,\frac12,1}_{\pm,\delta}$ on $[0,\delta]$, for $$\label{eq:deltaDef} 
\delta \approx \left( 