Complete symmetry group and nonlocal symmetries for some two-dimensional evolution equations

The complete symmetry group of an 1 + 1 evolution equation of maximal symmetry has been demonstrated to be represented by the six-dimensional Lie algebra of point symmetries sl ( 2 , R ) ⊕ s W , where W is the three-dimensional Heisenberg–Weyl algebra. We construct a complete symmetry group of a 1 + 2 evolution equation u t = ( F ( u ) u x ) y for some functions F using the point symmetries admitted by the equation. The 1 + 2 equation is not completely specifiable by point symmetries alone for some specific functions F. We make use of Ansätze already reported by Myeni and Leach [S.M. Myeni, P.G.L. Leach, Nonlocal symmetries and complete symmetry groups of evolution equations, J. Nonlinear Math. Phys. 13 (2006) 377–392] which provide a route to the determination of the required generic nonlocal symmetries necessary to supplement the point symmetries for the complete specification of these 1 + 2 evolution equations. Further we find that taking some suitable linear combination of Lie point symmetries helps to optimise the procedure of specifying the equation. A general result concerning the number of symmetries required to form a complete symmetry group of evolution is presented in the Conclusion.