T-II-1

Far from the beach , i.e., when the sea is deep, water waves travel on the surface of the sea and their shape is not deformed too much. When the waves approach the beach their shape begins to change and eventually breaks upon touching the seashore.

Here we are interested in studying a simple model that captures the essential features of this physical system. More precisely we will examine the Burgers equation, a non-linear equation introduced as a toy model for turbulence. this equation can be explicitly integrated and, in the limit of zero viscosity, its solution generally displays discontinuities called shocks.

Model 0 : Deep water

Consider the following equation for the shape of the wave:

$\partial _{t}u(x,t)=c\partial _{x}u(,t)$

where $u(x,t)$ is the height of the wave with respect to the unperturbed sea level, and c is constant. We will use the following initial condition

Model 1 : Approaching the sea shore

It is possible to show that, when the height of the wave is negligible as compared to the sea depth, the velocity of the wave does not depend on $u$. On the contrary, when approaching the sea shore this is not anymore true and the velocity becomes proportional to $u$: $v(u)\simeq c_{1}u+v_{0}$ where $c_{1}$ is a constant and $v_{0}$ is the velocity of the bottom of the wave. Let us take $c_{1}=1$ and write the equation for the wave motion in the frame of the bottom of the wave, which yields: $\partial _{t}u(x,t)=u(x,t)\partial _{x}u(x,t).$ This is the non linear equation originally introduced by Burgers. In Burgers' derivation a viscous term was also present $\partial _{t}u(x,t)=u(x,t)\partial _{x}u(x,t)+\nu \partial _{x}^{2}u(x,t),$

$\nu$ being the viscosity, and  Eq 1 is called the inviscid limit of Eq. 2.

Our strategy will be to investigate the properties of Eq1 by resorting to the method of characteristics. Moreover, we will integrate Eq 2 by using the Cole Hopf transformation, and then recover Eq.1 by taking the limit $\nu \to 0$.

The Method of characteristics

$u_{0}(x)={\begin{cases}1\\2\end{cases}}$