Second Order Nonlinear Differential Equations with Riccati-Type Solutions

Solutions to complex nonlinear ordinary differential equations (NODEs) are often obtained by seeking them in the form of series expansions of known solutions to an auxiliary equation. This paper presents an exhaustive study of a general class of second-order NODEs with polynomial coefficients and identifies the necessary, but not sufficient, conditions on the polynomial degrees of these coefficients for the equation to admit solutions expressible in terms of solutions of the Riccati equation. From this perspective, “Go/No-Go” criteria are formulated for numerous models of practical interest, such as the well-known Burgers, Korteweg–de Vries, Schrödinger, Klein–Gordon, Dodd–Bullough–Mikhailov, Fisher and Chaffee–Infante equations. As a case study, explicit solutions for a generalized model of the Hunter–Saxton equation are presented. The results summarized in the tables provide a useful criterion for assessing, prior to solving, whether a given second-order equation admits solutions in terms of an auxiliary Riccati equation, and indicate the expected form of such solutions.