0$ such that: given any $a\in (0, a_q]$, arbitrarily small $\varepsilon_0 > 0$ and arbitrarily large $N_0 > 0$, there exist $N>N_0$ and $w\in Y$ such that $\|w\|\leq \varepsilon_0$, $\|\clT^n(w)\|_{Y}\leq a$ for $01$, then $\phi$ is an unstable fixed point of $\clW$. \end{coro} \begin{proof} For $w\in U\equiv\{v-\phi;\ v\in \clO\}$, let us consider $\clT(w)\equiv \clW(w+\phi)-\phi$. Since $\phi$ is a fixed point of $\clW$, we have $\clT(0)=\clW(\phi)-\phi=0$ with $1<|\mu|\leq r(\clW'(\phi))$. By Taylor's formula, we obtain $ \clT(w)=\clT(0)+\clT'(0)w+ O(\|w\|_Y^2)=\clW'(\phi) (w)+O(\|w\|_Y^2)$ for all $\|w\|_Y<<1$. Therefore, by Remark~\ref{henry2}, there exists $\varepsilon_0>0$ such that for all $\eta>0$ and a large enough $N_0\in \bbN$, there exists $N>N_0$ and $v\in B(\phi;\eta)$ such that $\|\clW^N(v)-\phi\|_Y\geq \varepsilon_0.$ This finishes the proof. \end{proof} \begin{prop}\label{insta} The periodic solution $\varphi$ that is transversally (spectrally) unstable according to Theorem~\ref{mainT} is nonlinearly unstable. \end{prop} \begin{proof} By equation~\eqref{ZK}, we see that $u(x-ct,y,t)$ is a solution of the equation \begin{equation}\label{newgZK} u_t-cu_{\xi}+u^pu_{\xi}+(\Delta u)_{\xi}=0, \end{equation} where $\xi=x-ct$. In addition, the periodic wave $\varphi$ obtained in Section~\ref{sect2} is now an equilibrium solution of the equation~\eqref{newgZK}. Consider $G(u)=E(u)+cF(u)$, where $E$ and $F$ are given by~\eqref{conservada1}. We have that~\eqref{newgZK} can be rewritten as \begin{equation}\label{hamiltgBBM2} u_t=JG'(u), \end{equation} where $J=\partial_x$. Moreover, from~\eqref{hamiltgBBM2} the linearized equation at the equilibrium point $\varphi$ is $v_t=J(\clL-\partial_y^2)v$, where $\clL$ is the linear operator given by~\eqref{operator}. Let us consider $v(x,y,t)=e^{iky}w(x,t)$ in the linearized equation to obtain $w_t=J(\clL+k^2)w$. Define $\clW:X\rightarrow X$ as $\clW(u_0)=u_{u_0}(1)$, where $u_{u_0}(t)$ is the solution of~\eqref{newgZK} with initial data $u(x,y,0)=u_0(x,y)$ at $t=1$. For each $T>0$, function $\Upsilon:X\rightarrow C([0,T];X)$ is the data-solution map related to the equation~\eqref{newgZK} and by assumption (H1), $\Upsilon$ is smooth. Again by (H1), the uniqueness of solutions for the Cauchy problem~\eqref{cauchygZK} gives us that $\clW(\varphi)=\varphi$ and $\clW$ is a $C^2$ map defined in a neighbourhood of $\varphi$ (this fact follows from the translation in $x$ as a linear continuous map defined $X$). Moreover, for $h(x,y)=e^{iky}g(x)\in X$ we have $\clW'(\varphi)h=w_h(1)$, where $w_h(1)$ is the solution of the linear initial value problem \begin{equation}\label{linear1} \left\{ \begin{array}{lll} w_t=J\left(\clL+k^2\right)w\\ w(0)=h, \end{array}\right. \end{equation} evaluated at $t=1$. Then, using Theorem~\ref{mainT}, we obtain the existence of $\nu>0$, $k\neq0$ and $U\in X\backslash\{0\}$ such that $J(\clL+k^2)U=\nu U$. Hence, for $w_U(t)=e^{\nu t} U$ and $\alpha= e^\nu$, we obtain $\clW'(\varphi)U=w_U(1)=\alpha U$, that is, $\alpha\in\sigma(\clW'(\varphi))$. By Corollary~\ref{corohen}, we obtain the nonlinear instability in $Y$ of the periodic solution $\varphi$ that is transversally (spectrally) unstable according to the Theorem~\ref{mainT}. \end{proof} \section*{Declaration of interests} The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations. \printbibliography \end{document}