Let $$\begin{equation} u(x,t)= \begin{cases} \frac{x-2}{t+2}&;x>\xi(t)\\0&;x<\xi(t)\end{cases} \end{equation}$$ be a weak solution to $u_t+(\frac{u^2}{2})_x=0$ in $\mathbb R\times(0,\infty)$. The unknown shock curve $x=\xi(t)$ starts at the origin, then determine the shock curve and draw a picture with characteristics in the $xt$-plane.
we find characteristics curve by Rankine-Hugoniot, i'm not sure to sketch the diagram. Please, someone give me a detail.
This is the inviscid Burgers' equation. The Rankine-Hugoniot condition $$ \xi'(t) = \frac{1}{2}\left(\frac{\xi(t)-2}{t+2} + 0\right) $$ with initial position $\xi(0)=0$ gives the shock trajectory $\xi(t) = 2 - \sqrt{2t+4}.$ Characteristic curves are obtained via the method of characteristics with initial data $$ u(x,0) = \left\lbrace \begin{aligned} &0 & &\text{if}\quad x<0,\\ &x/2-1 & &\text{if}\quad x>0. \end{aligned} \right. $$ Here is a sketch of the $x$-$t$ plane so-obtained: