We have the following equation the nonhomogeneous Burgers equation with initial condition \begin{align*} uu_x +u_y&= 1\\ u(x,0)&=x\quad x\in\mathbb{R} \end{align*} corresponding to the manifold $\Gamma$ in the $xyz$ space given by \begin{equation} x = s,\quad y = 0,\quad z = s \end{equation} The characteristic differential equations \begin{equation*} \frac{dx}{dt}=z\quad \frac{dy}{dt}=1\quad \frac{dz}{dt}=1 \end{equation*} Integrating each of the expressions we have \begin{align*} \int dz&=\int dt\\ z&=t+f_1(s) \end{align*} combined with the initial condition for $t=0$ lead to the parametric representation \begin{align*} z&=t+s \end{align*} also \begin{align*} \int dy&=\int 1 dt\\ y&=t \end{align*} and \begin{align*} \int dx&=\int z dt\\ \int dx&=\int t+s dt\\ x&=\frac{t^2}{2}+st+s \end{align*} therefore the characteristics flowing out from $\Gamma$ are \begin{equation*} X(s,t)=\frac{t^2}{2}+st+s,\quad Y(s,t)=t, \quad Z(s,t)=t+s \end{equation*} we find the solution \begin{equation*} u=y+\frac{2x-y^2}{2y+2} \end{equation*}
Weak (or Integral) Solutions
In this case the characteristics are curves that are given as follows \begin{equation*} X(s,t)=\frac{t^2}{2}+st+s \end{equation*} that this characteristic presents shock.
I have several questions the first one is how this characteristic presents shock.
a) How find $y_s$ (breaking time) and the location $x_s$?
b) In this case, how is the Rankine-Hugoniot study done?
Your answers would be of great help. Thank you very much. I remain attentive to your answer.