Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 32E / b

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.