Principle of structural analogy of solutions and its application to nonlinear PDEs and delay PDEs (2405.20474v1)

Abstract: Using the principle of structural analogy of solutions, approaches have been developed for constructing exact solutions of complex nonlinear PDEs, including PDEs with delay, based on the use of special solutions to auxiliary simpler related equations. It is shown that to obtain exact solutions of nonlinear non-autonomous PDEs, the coefficients of which depend on time, it is possible to use generalized and functional separable solutions of simpler autonomous PDEs, the coefficients of which do not depend on time. Specific examples of constructing exact solutions to nonlinear PDEs, the coefficients of which depend arbitrarily on time, are considered. It has been discovered that generalized and functional separable solutions of nonlinear PDEs with constant delay can be used to construct exact solutions of more complex nonlinear PDEs with variable delay of general form. A number of nonlinear reaction-diffusion type PDEs with variable delay are described, which allow exact solutions with generalized separation of variables.