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:ux=f(x)F(u)+εF(u)exp(−∫u1F(z)dz)}. The invariant set and the exact solutions to the nonlinear diffusion equation ut=(A(u)ux)x+Q(x,u)ux+P(x,u)ut=(A(u)ux)x+Q(x,u)ux+P(x,u), are discussed. It is shown that there exists a class of solutions to the equation that belong to the invariant set EE. The invariant set is also used to construct exact solutions to some other nonlinear evolution equations.