Higher-order Abel equations: Lagrangian formalism, first integrals and Darboux polynomials

Abstract

A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and it is proved the existence of two alternative Lagrangian formulations, both Lagrangians being of a non-natural class (neither potential nor kinetic term). These higher-order Abel equations are studied by means of their Darboux polynomials and Jacobi multipliers. In all the cases a family of constants of the motion is explicitly obtained. The general n-dimensional case is also studied.

Nonlocal symmetries of Riccati and Abel chains and their similarity reductions

“…The second ODE in this family is the modified Emden equation 9 The higher order Abel equations/Abel chain has the following form of differential equations, 4,8 namely,…”

“…10 The properties exhibited by Riccati and Abel chains make them interesting from both physical and mathematical points of view. 4 Very recently, n-dimensional integrable generalizations of both the chains have also been proposed.…”

Symmetries of nonlinear ordinary differential equations: The modified Emden equation as a case study

“…In a similar way one can fix the λ-symmetries for the remaining vector fields. The resultant expressions are given in the following Table. One can verify that the functions ξ, η and λ satisfy the invariance condition (54). Since we are dealing with a second-order ODE two different λ functions are sufficient to generate two independent first integrals and hence the general solution.…”

“…The example which we have chosen to illustrate the symmetry methods is the modified Emden equation (MEE) [48,49,50,51,52,53,54,55] …”

Solution of Higher-Order Abel Equations by Differential Transform Method

“…Carinena et al 30 in their recent paper have studied the existence of a Lagrangian formulation and have obtained some constants of motion particularly for the secondorder Abel equation. They have also established the relationship of the second-order Abel equation with the theory of Darboux polynomials and Jacobi multipliers.…”