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 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.