Overdetermined Systems of ODEs with Parameters and Their Applications: The Method of Differential Constraints and the Generalized Separation of Variables in PDEs

Abstract

Various situations are described where the construction of exact solutions of nonlinear ordinary and partial differential equations leads to overdetermined systems of ODEs with parameters that are not included in the original differential equations. A non-classical problem for ordinary differential equations with parameters is formulated and the concept of the conditional capacity of an exact solution is introduced. The method for investigating overdetermined systems of two ODEs of any order on consistency, which eventually leads to algebraic equations with parameters, is presented. A general description of the method of differential constraints with respect to ordinary differential equations is given and many specific examples of applying this method for obtaining exact solutions are considered. It is shown that the use of the splitting method (and also the method based on the use of invariant subspaces of nonlinear operators) for constructing exact generalized separable solutions of nonlinear PDEs can lead to overdetermined systems of ODEs with parameters. Several nonlinear partial differential equations (including a delay PDE) of higher orders are considered, and their exact solutions are found by analyzing the corresponding overdetermined ODE systems.