Speaker
Didier Pilod, University of Bergen
Abstract
This talk is concerned with the Zakharov-Kuznetsov equation
\begin{equation} \label{ZK}
\partial_tu+\partial_x(\Delta u+u^2)=0 ,\end{equation}
where \(u=u(t,x,y)\) is a real valued function and \((x,y) \in \mathbb R^2\). This equation appears as an asymptotic model to describe the propagation of ionic-acoustic waves in a uniformed magnetized plasma. Moreover, it is a natural mathematical generalization of the Korteweg-de Vries equation to higher dimension.
The Zakharov-Kuznetsov admits a family of solitary wave solutions propagating at constant speed in the horizontal direction.
They are of the form
\begin{equation}
u_c(t,x,y)=Q_c(x-ct,y), \quad \text{for} \ c>0 ,
\end{equation}
where \(Q_c(x,y)=cQ(\sqrt{c}x,\sqrt{c}y)\) and \(Q\) is a ground state of -\(\Delta Q+Q-Q^2=0\).
In this talk, we will study the collision of two solitary waves \(Q_{c_1}\), \(Q_{c_2}\) with nearly equal size, i.e. \(\mu_0=c_2-c_1 >0\) is small.
This is a joint work with Frédéric Valet (UiB).