Mp. Edwards et P. Broadbridge, EXCEPTIONAL SYMMETRY REDUCTIONS OF BURGERS-EQUATION IN 2 AND 3 SPATIAL DIMENSIONS, Zeitschrift fur angewandte Mathematik und Physik, 46(4), 1995, pp. 595-622
Lie point symmetry analysis of the general class of nonlinear diffusio
n-convection equations in two and three dimensions has shown that only
for Burgers' equation (that is D(u) = const, K(u) = quadratic) is a f
ull symmetry reduction to an ordinary differential equation possible.
The optimal system of symmetry operators is determined to ensure that
a minimal complete set of reductions is obtained. For each reduced par
tial differential equation, classical Lie group analysis has been perf
ormed and further reductions obtained. In this manner, all possible re
ductions to an ordinary differential equation are found, leading to ex
act solutions to both the two and three dimensional Burgers' equation.