Travelling wave solution of a PDE

A function $u (x, t)$ is a travelling wave solution for a given PDE if it has the form

u (x, t) = f (x - c t)

It is a fact that so many PDEs have solutions of this type, mainly the wave equation.

Example: The equation $u_{t} + c u_{x} = 0$ admits any function of this type as a solution, as it is easy to check. This is the transport equation.
Example of non-existence: there are equations which don't have solutions of this kind. For example, the diffusion equation

u_{t} = D u_{x x}

They are the similarity solutions associated to the symmetry

c \partial_{x} + \partial_{t}