Factorization conditions for nonlinear second-order differential equations

TL;DR

This paper explores new factorization conditions for nonlinear second-order differential equations with polynomial nonlinearities, using commuting and non-commuting approaches, and applies these to classical equations like Fisher and FitzHugh-Nagumo.

Contribution

It introduces two novel approaches to factorization conditions, expanding the class of solvable nonlinear second-order differential equations.

Findings

Commuting factorization brackets lead to Ermakov-Pinney type equations.

Non-commuting approach yields different nonlinear force terms.

Applications demonstrated on Fisher and FitzHugh-Nagumo equations.

Abstract

For the case of nonlinear second-order differential equations with a constant coefficient of the first derivative term and polynomial nonlinearities, the factorization conditions of Rosu and Cornejo-Perez are approached in two ways: (i) by commuting the subindices of the factorization functions in the two factorization conditions and (ii) by leaving invariant only the first factorization condition achieved by using monomials or polynomial sequences. For the first case, the factorization brackets commute and the generated equations are only equations of Ermakov-Pinney type. The second modification is non commuting, leading to nonlinear equations with different nonlinear force terms, but the same first-order part as the initially factored equation. It is illustrated for monomials with the examples of the generalized Fisher and FitzHugh-Nagumo initial equations. A polynomial sequence example is also included.