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.

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