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Let $ϵ$ be the group parameter. The infinitesimal generator of a Lie point symmetryhas flow equationsTherefore it generates the scaling group

Here ${\xi }^t=t$, ${\xi }^x=x$, and $\eta =0$. The total-derivative formulas for the second prolongation of a Lie point symmetry giveandThusFor the wave equation expression $F=u_{tt}-u_{xx}$,It therefore vanishes whenever $F=0$, proving that $V$ generates the simultaneous spacetime scaling symmetry of the wave equation.

The independent invariant of the scaling orbits is the similarity variablewhile $u$ itself is invariant. Seek a group-invariant solution $u(t,x)=f(z)$. Direct partial differentiation givesThe wave equation reduces toHence, on any interval avoiding $z=\pm 1$,

Forthe flow equations arewith $(T,X,U)=(t,x,u)$ at $ϵ=0$. Solving the first two gives, with $d=1-4ϵt$,Integrating the last equation then gives the one-parameter groupdefined locally where $1-4ϵt>0$.

To verify the symmetry infinitesimally, useThe needed coefficients of the second prolongation of a Lie point symmetry areandFor the potential Burgers equation written aswe obtainThus the prolonged generator is tangent to the solution manifold $F=0$, and the finite transformations above form the projective Lie symmetry of the potential Burgers equation.