Transport equation

Also known as advection equation and first order linear wave equation. Related: wave equation.
It is the simplest PDE exhibiting travelling wave solutions (it is said in this video). We want a function $u(x,t)$

with $c\in \mathbb{R}$.

In the video it is said that the characteristics lines are those curves in space $(x,t)$ such that $u$ is constant along them: $u(\gamma (t))=cte$. That is, the red lines below:

This is probably related to similarity solutions and to the method of characteristics.