Heuristic analysis of the complete symmetry group and nonlocal symmetries for some nonlinear evolution equations

The complete symmetry group of a 1 + 1 evolution equation has been demonstrated to be represented by the six-dimensional Lie algebra of point symmetries sl(2, R) ⊕sW, where W is the three-dimensional Heisenberg–Weyl algebra. We construct a complete symmetry group of a nonlinear heat equation ut= F(ux)uxx for some smooth functions F, using the point symmetries admitted by each equation. The nonlinear heat equation is not specifiable by point symmetries alone even when the number of symmetries is 6. We report Ansatze which provide a route to the determination of the required nonlocal symmetry necessary to supplement the point symmetries for the complete specification of these nonlinear 1 + 1 evolution equations. The nonlocal symmetry immediately realized is said to be generic to a class of equations as it gives a specific structure to an equation. Copyright © 2007 John Wiley & Sons, Ltd.