Title: Concentration-Compactness and the long time dynamics of the
generalized Korteweg-de Vries equation
Abstract
We consider the Cauchy problem associated with the generalized KdV
equation:
\begin{equation}
\label{kdv} \left.\begin{array}{ll} & u_t+u_{xxx}+(f(u))_{x}=0, \
\ \ (t,x)\in{\mathbb{R}}^+\times{\mathbb{R}},\\
&u(0,\cdot)=u_0. \end{array}\right\}
\end{equation}
This includes the standard KdV ($f(u)=u^2)$) and modified KdV
($f(u)=u^2u$). It is well known that both these equations are
completely integrable. The main question is weather (\ref{kdv})
retains any of the striking wave-like phenomena observed for these two
special cases using techniques from inverse scattering . This includes the
emergence of a train of solitary waves from arbitrary initial data,
and the elastic collision of solitary waves. In this talk we will give
an overview of recent results and present some of the tools used in analyzing
(\ref{kdv}) outside the realm of completely integrable equations. We
will focus on the application of P.-L. Lions'
Concentration-Compactness principle, and discuss how, due to the presence
of certain conserved quantities, the principle is a very natural
setting for studying the long time dynamics of (\ref{kdv}). As an
example, we use the principle to obtain the stability of traveling waves.