have $v_{-\alpha}=\overline{v_{\alpha}}$. Moreover, let the corresponding profile $f$ be$$\label{deff}f_{\nu}(t)=e^{{i}t\Lambda_{\nu}}v_{\nu}(t),\qquad0\leq t\leq T.$$

\[loc\] Suppose $g,h:\mathbb{R}^{3}\to\mathbb{R}^{d}$ are such that $\|(g,\partial_{x}g,h)\|_{\mathcal{H}^{N}}\leq\varepsilon_{0}$, then there exists a unique solution $u$ to (\[nlkg\]) such that $$u\in C_{t}^{1}\mathcal{H}_{x}^{N}([0,1]\times\mathbb{R}^{3}\to\mathbb{R}^{d}),\quad\partial_{x}u\in C_{t}^{0}\mathcal{H}_{x}^{N}([0,1]\times\mathbb{R}^{3}\to\mathbb{R}^{d});\quad u(0)=g,\partial_{t}u(0)=h.$$ Moreover, if $\|(g,\partial_{x}g,h)\|_{Z}\leq\varepsilon_{0}$, then we have $f(t)\in C([0,1]\to Z)$, where $f(t)=(f_{\nu}(t))$ is defined as in (\[deff\]) above.

This is proved, with slightly different parameters, in [@IP12], Proposition 2.1 and 2.4; the proof in our case is basically the same.

With Proposition \[loc\], we can reduce the proof of Theorem \[mainkg\] to the following a priori estimate.

\[bootstrap\] Suppose $u$ is a solution to (\[nlkg\]) on a time interval $[0,T]$ with initial data $u(0)=g$ and $u_{t}(0)=h$ such that $$u\in C_{t}^{1}\mathcal{H}_{x}^{N}([0,T]\times\mathbb{R}^{3}\to\mathbb{R}^{d}),\quad\partial_{x}u\in C_{t}^{0}\mathcal{H}_{x}^{N}([0,T]\times\mathbb{R}^{3}\to\mathbb{R}^{d}),$$ and let $f(t)$ be defined accordingly. Assume $$\|(g,\partial_{x}g,h)\|_{X}\leq\varepsilon\leq\varepsilon_{0},\qquad \sup_{0\leq t\leq T}\|f(t)\|_{X}\leq\varepsilon_{1}\ll 1,$$ then we have $$\sup_{0\leq t\leq T}\|f(t)\|_{X}\lesssim \varepsilon_{1}^{3/2}+\varepsilon.$$

Control of energy
-----------------

From now on we will fix a solution $u$ as described in Proposition \[bootstrap\], and the corresponding $f$; in this section we will recover the energy bounds.

\[econtrol\] We have $$\label{resenergy}\sup_{0\leq t\leq T}\sup_{|\mu|\leq N}\|\Gamma^{\mu}f(t)\|_{L^{2}}\lesssim \varepsilon_{1}^{3/2}+\varepsilon.$$

Recall that$$\|\Gamma^{\mu}f(t)\|_{L^{2}}\sim\|\Gamma^{\mu}u(t)\|_{L^{2}}+\|\Gamma^{\mu}(\partial_{x},\partial_{t})u(t)\|_{L^{2}},$$the proof is basically the same as in [@IP12]; we will present it here since the norms involved are different. Define the energy$$\begin{aligned}
\mathcal{E}(t)&=&\sum_{|\mu|\leq N}\int_{\mathbb{R}^{3}}\bigg(\sum_{\alpha=1}^{d}(|\partial_{t}\Gamma^{\mu}u_{\alpha}|^{2}+b_{\alpha}^{2}|\Gamma^{\mu}u_{\alpha}|^{2}+c_{\alpha}^{2}|\nabla\Gamma^{\mu}u_{\alpha}|^{2})+\\
&+&\sum_{\alpha,\beta=1}^{d}\sum_{j,k=1}^{3}S_{\alpha\beta}^{jk}(u,\partial u)\partial_{j}\Gamma^{\mu}u_{\alpha}\cdot\partial_{k}\Gamma^{\mu}u_{\beta}\bigg)\,\mathrm{d}x\nonumber,\end{aligned}$$ where 