Nonlinear wave equations and compatibility conditions for polynomial differential relations

Background. The connection of the compatibility conditions for nonlinear differential relations of polynomial type with nonlinear wave and diffusion equations is considered. This approach is a variant of the method of nonlinear functional substitutions, which is one of the methods for constructing exact solutions to nonlinear partial differential equations. Materials and methods. The main method used in the work is the method of non-linear functional substitutions, which is a development of the method of functional substitutions, which was previously used to construct solutions to Burgers-type equations. The study considers a variant of the method of nonlinear functional substitutions with basic relations that are polynomial in the auxiliary function. Results. The conditions for the compatibility of basic relations with polynomial basic functions are completely analyzed. New exact solutions of equations of the diffusion type are found and the methodology for applying the method in practice is indicated. As examples, a new chain of Burgers equations with nonlinear sources is given. A connection between the compatibility conditions and the Lax representation is established. Conclusions. The developed approach makes it possible to construct exact solutions of new nonlinear equations.