Time: 9h30, Tuesday, February 16, 2016
Location: Room 4, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi
Abstract:  The (2-D) (scalar) Zakharov system has the form
\[
 \left \{
   \begin{aligned}
    & i \partial_t u + \Delta u = n u \\
    & \dfrac{1}{\lambda^2} \partial_t^2 n - \Delta n = \Delta |u|^2.
   \end{aligned}
\right.
\]
Where $ (x,y,t) \in \R^2 \times \R $, $\lambda$ is a fixed real number, $u$ is a complex valued function and $n$ is  a real function, with initial data $(u_0, n_0, n_1)$.
The Zakharov system describe the propagation of Langmuir waves in plasma and was studied by many authors and various methods. In this talk, I represent some classical result about the Cauchy problem for the Zakharov system and, then, present our recent result on the Cauchy problem for its perturbation on the background of the line soliton \footnote{The 1-d soliton of the 1-d focusing nonlinear Schr\"odinger equation.} . One can consider it as the first step to study the transverse stability (or instability) of the line soliton.