Scaling symmetry

A scaling symmetry acts by weighted dilations of the independent and dependent variables. Its infinitesimal generator is a weighted sum of Euler vector fields such as $x{\partial }_x+pu{\partial }_u$.

A scaling is a symmetry when every term of the equation transforms with the same weight.

The flow of $V=t{\partial }_t+x{\partial }_x$ is $(t,x,u)\mapsto (e^{ϵ}t,e^{ϵ}x,u)$. Its second prolongation scales both $u_{tt}$ and $u_{xx}$ with weight minus two, so it preserves $u_{tt}-u_{xx}=0$.