Burgers PDE with piecewise constant initial condition Let's deal with this Burgers PDE:
$$\left\lbrace
\begin{aligned}
&u_t + uu_x = 0, \quad x\in \Bbb{R}, t>0\\
&u(x,0) = \varphi(x)
\end{aligned}
\right.
$$
where
$$\varphi(x) = \left\lbrace
\begin{aligned}
&c_1, \quad x\leq \alpha\\
&c_2, \quad \alpha\leq x\leq \beta, \quad c_1> c_2> c_3 .\\
&c_3, \quad x\geq \beta
\end{aligned}
\right.
$$
You can see the characteristic curves below:


*

*I believe the breaking time is $\boldsymbol{t_b=0}$, because it is the $\inf$ of $t$-coordinates of the intersection points of characteristics curves. Am I right?

*How can I find the weak solution? How can I find the discontinuity line? At $t=0$ we have $x=\alpha$ and $x=\beta$ at the intersection points.

Thanks, in advance!
 A: Apply the techniques in this post. Because $c_1>c_2>c_3$, the solution for small time will be shock waves whose speed is deduced from the Rankine-Hugoniot condition
$$
s_{12} = \tfrac12(c_1+c_2)> c_2, \qquad s_{23} = \tfrac12(c_2+c_3) < c_2.
$$
Hence, for small times,
$$
u(x,t) = \left\lbrace
\begin{aligned}
&c_1 , & & x\leq \alpha + s_{12} t\\
&c_2 , & & \alpha + s_{12} t \leq x\leq \beta + s_{23} t\\
&c_3 , & & x\geq \beta + s_{23} t
\end{aligned}
\right.
$$
Waves of amplitude $c_1$, $c_3$ will interact at some time $t^*>0$ if
$$
\alpha + s_{12} t^* = \beta + s_{23} t^* = x^*,
$$
i.e., at the time
$$
t^* = \frac{\beta -\alpha}{s_{12}-s_{23}} > 0
$$
and position
$$
x^* = \frac{s_{12}\beta-s_{23}\alpha}{s_{12}-s_{23}} .
$$
This is very much possible since $\beta -\alpha \geq 0$ and $s_{12}-s_{23}>0$. The resulting shock wave has Rankine-Hugoniot speed $s_{13} = \frac12(c_1+c_3)$, and we have
$$
u(x,t) = \left\lbrace
\begin{aligned}
&c_1 , & & x\leq x^* + s_{13} t\\
&c_3 , & & x\geq x^* + s_{13} t
\end{aligned}
\right.
$$
for $t>t^*$.
