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Projective Lie symmetry of the potential Burgers equation

The vector field

V = 4 t^{2} \partial_{t} + 4 t x \partial_{x} - (x^{2} + 2 t) \partial_{u}

generates the local transformations

T = \frac{t}{1 - 4 ϵ t}, X = \frac{x}{1 - 4 ϵ t}, U = u - \frac{ϵ x ^{2}}{1 - 4 ϵ t} + \frac{1}{2} lo g (1 - 4 ϵ t) .

Its second prolongation sends

u_{t} - u_{xx} - u_{x}^{2}

to

- 8 t

times that expression, so it is a Lie point symmetry of the potential Burgers equation.

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Solution