Conditional Lie–Bäcklund symmetry, second-order differential constraint and direct reduction of diffusion systems

The equivalence relation between conditional Lie–Bäcklund symmetry, higher-order differential constraint and direct reduction is presented in the case of evolution systems. The second-order nonlinear conditional Lie–Bäcklund symmetries and corresponding induced differential constraints admitted by nonlinear diffusion systems are identified as an application of the conditional Lie–Bäcklund symmetry of evolution systems. Consequently, the corresponding symmetry reductions are obtained due to the compatibility of the resulting differential constraints and the governing systems.