| Abstract: |
| Considered here are two unidirectional water wave models for small amplitude long waves on the surface of an ideal fluid
$\begin{equation}
\eta_t + \eta_x + \frac34 \alpha (\eta^2)_x - \frac16\beta \eta_{xxt} \, = \, 0,
\end{equation}$
and the higher-order model equation
$\begin{equation}\begin{split}
\eta_t+\eta_x-\gamma_1\beta{\eta}_{xxt}+\gamma_2\beta\eta_{xxx}+\delta_1\beta^2{\eta}_{xxxxt}
+\delta_2\beta^2\eta_{xxxxx}
\
+ \frac34\alpha(\eta^2)_x+
\alpha\beta\Big(\gamma (\eta^2)_{xx}-\frac7{48}\eta_x^2\Big)_x-\frac18\alpha^2(\eta^3)_x=0,
\end{split}\end{equation}$
where $\eta=\eta(x,t)$, $x\in\mathbb R$ and $t\geq 0$, is the deviation of the free surface from its rest position at the point corresponding to $x$ at time $t$.
$\alpha, \beta $ $\gamma_1, \gamma_2, \delta_1, \delta_2$, $\gamma $ are physical parameters.
In this talk, we discuss well-posedness issues when the initial dada is non-localized. |
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