I would like to collect some test-problems for nonlinear hyperbolic PDEs (Euler Equations, Shallow Water Equations, Ideal MHD, Acoustic Perturbation, ...) for which analytical solutions are known.
A famous example of this is the isentropic vortex advection by Shu for the 2D Euler equations. For the Euler equations, some more examples are also given in Chapter 7.13 of this book. In 1D, often also solutions (at least for Riemann problems) can be constructed. Examples of this include Burger's Equation and the LWR traffic model.