Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation

In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution.