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$ u_{t}+u_{x}+u u_{x}-u_{x x t} = 0 $

was derived as a model for the unidirectional propagation of long-crested, surface water waves. It arises in other contexts as well, and is generally understood as an alternative to the Korteweg-de Vries equation. Considered here is the initial-value problem wherein $u$ is specified everywhere at a given time $t = 0$, say, and inquiry is then made into its further development for $t>0$. It is proven that this initial-value problem is globally well posed in the $L^2$-based Sobolev class $H^s$ if $s \geq 0$. Moreover, the map that associates the relevant solution to given initial data is shown to be smooth. On the other hand, if $s < 0$, it is demonstrated that the correspondence between initial data and putative solutions cannot be even of class $C^2$. Hence, it is concluded that the BBM equation cannot be solved by iteration of a bounded mapping leading to a fixed point in $H^s$-based spaces for $s < 0$. One is thus led to surmise that the initial-value problem for the BBM equation is not even locally well posed in $H^s$ for negative values of $s$.

$ u_t+u_x+g(u)_x+Lu_t=0, \qquad x\in\mathbb R,\quad t>0,$

$u(x,0)=u_0(x), \qquad x\in\mathbb R, (0.1)$

where $u=u(x,t)$ is a real-valued function, $L$ is a Fourier multiplier operator with real symbol $\alpha(\xi),$ say, and $g$ is a smooth, real-valued function of a real variable. Equations of this form arise as models of wave propagation in a variety of physical contexts. Here, fundamental issues of local and global well-posedness are established for $L_p$, $H^s$ and bore-like or kink-like initial data. In the special case where $\alpha(\xi)=|\xi|^{r}$ wherein $r>1$ and $g(u)=1/2u^2,$ (0.1) is globally well-posed in time if $s$ and $r$ satisfy a simple algebraic relation.

The focus of the session was asymptotic models of physical phenomena. This is a large subject, and one special session cannot hope to do it justice. Consequently, some focal point was required, and in the event, most of the lectures in the session were centered upon nonlinear wave equations arising in plasma physics and fluid mechanics. This is also the subject around which most of this special issue turns.

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$ u_t- $ u_{xxt}$ +u_x-$uu_{t}$ +u_x\int_x^{+\infty}u_tdx'=0, $ (1)

was developed by Hirota and Satsuma as an approximate model for unidirectional propagation of long-crested water waves. It possesses solitary-wave solutions just as do the related Korteweg-de Vries and Benjamin-Bona-Mahony equations. Using the recently developed theory for the initial-value problem for (1) and an analysis of an associated Liapunov functional, nonlinear stability of these solitary waves is established.

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