Extended rotation and scaling groups for nonlinear diffusion equations

In this paper, we introduce a new nonlinear extension to the rotation and scaling groups, which is described by the invariant set E = {u : u_x = f (x) F (u) + ε F (u) exp (- ∫^u frac(1, F (z)) d z)}. The invariant set and the exact solutions to the nonlinear diffusion equation u_t = (A (u) u_x)_x + Q (x, u) u_x + P (x, u), are discussed. It is shown that there exists a class of solutions to the equation that belong to the invariant set E. The invariant set is also used to construct exact solutions to some other nonlinear evolution equations.