Another difficulty can occur with nonlinear equations having discontinuous initial or boundary data.
Example. Consider the equation
subject to the initial condition
The characteristic curves are the following: on and on . Because is constant along characteristics. the data are carried into the region along characteristics with speed , and the data are carried into the region along vertical (speed ) characteristics.
There is a region void of the characteristics. In this case there is a continuous solution that connects the solution ahead to the solution behind. We simply insert characteristics (straight lines in this case) passing through the origin into the void in such a way that is constant on the characteristics and varies continuously from to along these characteristics. In other words, along the characteristic , , take . Consequently, the solution to the Riemann problem is
A solution of this type is called a centered expansion wave, or a fan; other terms are release wave or rarefaction wave. The idea is that the wave spreads as time increases.
Source: J.D. Logan, An introduction to nonlinear partial differential equations, 2nd, 2008; Section 3.1.