# Bờ-nốc bình dân

Giản dị như gió, nhẹ nhàng như mây…

## Thesis

My thesis, realized at the university of Nice Sophia-Antipolis under supervision of very kindly-hearted and reponsible professor Gilles LEBEAU, dealt with strongly non-linear waves quation, is concentrated on the following equation

$\left\{ \begin{array}{l}h^2(\partial^2_t-\partial^2_x)u + \frac{\partial F}{\partial u}(x,u) = \mathcal{O}(h^{\infty})\\ u_0(x)=u(0,x),\ u_1(x)=\partial_t(0,x) \in\mathcal{C}^{\infty} \end{array}\right. (1)$

where $h$ is very small parameter introduced that make (1) a very high-frequency type. The data are smooth that possess some asymptotic developments. We consider this problem on short interval of time depending $h$

A well-known result for this kind of equation is that, one can contruct a formal optical solution on fixed interval of time, independent of $h$ (1). Moreover, this one is unique. However we don’t know yet whether the exact solution exists. This problem is still open so far.

The work of my thesis is to prove that under some contexts, the optical solution for $(1)$ is completely unstable. In other words, I demonstrated that the difference between the optical solution and the exact one, if this exists, becomes significant after a very short of time. Hence, optical solution doesn’t allow us to calculate the exact one.

Open problem issued form my thesis is the equation of the potential function $F(u)=F(t_0,u)$ for which the unstability of the Hill equation holds

$2(a-F)F^{(4)} - F^{(1)}.F^{(3)}= -3(F^{(2)})^2 + b.F^{(2)}+c$

where $a,b,c$ are some constants and $F^{(n)}$ denotes $F$‘s $n^{th}$ derivative respectively

If ever you find it interesting, please take a look to the full report of my thesis here