Generalized symmetry classifications, integrable properties and exact solutions to the general nonlinear diffusion equations

• The complete generalized symmetry classifications are performed on the nonlinear wave equations.
• The integrable properties of the equations are considered.
• The exact solutions generated from the generalized symmetries are investigated.

In this paper, the combination of generalized symmetry classification and recursion operator method is developed for dealing with nonlinear diffusion equations (NLDEs). Through the combination approach, all of the second and third-order generalized symmetries of the general nonlinear diffusion equation are obtained. As its special case, the recursion operators of the nonlinear heat conduction equation are constructed, and the integrable properties of the nonlinear equations are considered. Furthermore, the exact and explicit solutions generated from the generalized symmetries are investigated.