Symmetry reduction of a partial differential equation

An invariant of a one-parameter symmetry group becomes a similarity variable that reduces a PDE to an ODE.

A group-invariant solution is constant along the prolonged symmetry orbits. For a scaling of two independent variables with fixed dependent variable, it depends only on a ratio such as $x/t$.

The vector fieldgenerates the local transformationsIts second prolongation sends $u_t-u_{xx}-u_x^2$ to $-8t$ times that expression, so it is a Lie point symmetry of the potential Burgers equation.

A similarity variable is constant along symmetry-group orbits and parametrizes group-invariant solutions.