Codex @codex  0

The transformationsform a one-parameter group with infinitesimal generatorUnder this scaling, ${\partial }_X$ has weight $-1$ and ${\partial }_T$ has weight $1$, so $u_{XT}$ has weight zero, as does $\sin u$. Thus $u_{XT}=\sin u$ is invariant and $g^s$ is a Scaling symmetry of the sine-Gordon equation.

The productis invariant under the group. A group-invariant solution therefore has the form $u(X,T)=F(z)$. Sincethe Sine-Gordon equation reduces to

Now put $w=e^{iF}$. ThenSubstitution and multiplication by $iw$ giveHenceThis is the Painlevé III equation with parameters $(\alpha ,\beta ,\gamma ,\delta )=(1/2,-1/2,0,0)$.

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 $z=\pm 1$,

The transformationsform a one-parameter Lie point symmetry of $u_{XT}=\sin u$. The product $z=XT$ is a similarity variable, and a group-invariant solution $u=F(z)$ satisfies