Generalized equations with a given soliton type solution. The example of the Chafee-Infante model

Abstract The paper proposes a procedure for extending a nonlinear ordinary differential equation (NODE) to other equivalent equations, in the sense that they accept a certain predefined solution, specific to the initially considered equation. We look for such extensions in the class of reaction-diffusion equations, second-order NODEs with polynomial coefficients. The Chafee-Infante (CI) equation is chosen as a specific representative of this class and various generalized CI equations with higher-order nonlinear coefficients are highlighted. The predefined solution that had to be satisfied by both the standard CI equation and its extensions was chosen as a first-order polynomial in the solutions of the generalized Riccati equation. The procedure we propose provides a criterion to establish ab initio whether a given equation can be solved using Riccati as an auxiliary equation. It can also be useful in defining classes of equivalent equations in terms of the solutions they accept.