Let us consider the initial value problem
ubject to the initial condition
Note that the characteristics emanate with speed from the axis.
That means the characteristics starting from are given as the following
Actually, I have not shown this fact before, if I have time then I will post something regarding to the characteristics of nonlinear equations. Of course the shock must form . The shock must start with discontinuity points. Clearly, is discontinuous at and . However, at , the characteristics do not intersect each other, therefore the shock must start at . By the Rankine-Hugoniot condition, the speed of the shock is
where is the value of ahead of the shock and is the value of behind the shock. Clearly, a shock must from at with speed
and propagate until time . To continue the shock beyond , we must know the solution ahead of the shock. Therefore, we introduce an expansion wave
The shock beyond according to the jump condition has speed
To find the shock path we note that , so the above equation is a first order ordinary differential equation for the shock path. Obviously, this ODE is given as
which is also a linear equation. The initial condition is at where the shock starts beyond . The solution is
The above shock will propagate until and . When , the new jump condition is
Therefore the shock is a straight line with speed for and its equation is
Source: J.D. Logan, An introduction to nonlinear partial differential equations, 2nd, 2008; Section 3.1.