such that we have scattering in slightly weaker spaces. $$\label{scat}\lim_{t\to\pm\infty}\big(\|u(t)-w^{\pm}(t)\|_{\mathcal{H}^{N-1}}+\|(\partial_{x},\partial_{t})(u(t)-w^{\pm}(t))\|_{\mathcal{H}^{N-1}}\big)=0.$$

\[remark9\] Note that $\varepsilon_{0}$ depends on $(b_{\alpha},c_{\alpha})$ in a way that is not necessarily continuous. This is because the proof relies heavily on the structure of the nonlinear phase function$$\Phi(\xi,\eta)=\sqrt{c_{\alpha}^{2}|\xi|^2+b_{\alpha}^2}\pm\sqrt{c_{\beta}^{2}|\xi-\eta|^2+b_{\beta}^2}\pm\sqrt{c_{\gamma}^{2}|\eta|^2+b_{\gamma}^2},$$ which could be sensitive to small changes in $c_{\alpha}$ and $b_{\alpha}$. For example, when $b_{\alpha}+b_{\beta}\neq b_{\gamma}$, the constant $K_{0}$ in Section \[choiceof\] (which ultimately determines $\varepsilon_0$) will depend on the value of $\theta=|b_{\alpha}+b_{\beta}-b_{\gamma}|$, and our argument does not rule out the theoretical possibility that $K_{0}\to\infty$ when $\theta\to 0$ (on the other hand, if $b_{\alpha}+b_{\beta}= b_{\gamma}$ then we have a different argument that gives a finite $K_0$).

We believe that this annoyance is of technical nature, and can be avoided by more careful analysis of $\Phi$. However, since this is not very relevant to the main idea of this paper, and will make the proof unnecessarily long, we have chosen to state Theorem \[mainkg\] as it is.

Notations and choice of parameters {#choice}
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### Notations {#notatkg}

We fix an even smooth function $\varphi:\mathbb{R}\to[0,1]$ that is supported in $[-8/5,8/5]$ and equals $1$ in $[-5/4,5/4]$. It is well-known (see for example [@Ro]) that we can assume $\varphi\in\mathcal{G}_{4}(\mathbb{R})$, where $\mathcal{G}_{\sigma}$ is the Gevrey 