A nonlocal connection between certain linear and nonlinear ordinary differential equations : Extension to coupled equations

TL;DR

This paper introduces a novel nonlocal transformation approach to identify and solve integrable coupled nonlinear ODEs of arbitrary order, reducing complex systems to simpler first order equations and providing explicit solutions.

Contribution

It presents a new nonlocal connection method that extends to coupled equations, enabling the reduction of complex systems to single first order ODEs and deriving explicit solutions.

Findings

Identified two classes of integrable coupled nonlinear ODEs of arbitrary order.

Reduced solving these systems to a single first order nonlinear ODE.

Derived explicit solutions for various classes including modified Emden and Chazy equations.

Abstract

Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and unearth two classes of integrable coupled nonlinear ODEs of arbitrary order. To achieve these goals we introduce suitable nonlocal transformations in certain linear ODEs and generate the coupled nonlinear ODEs. In particular, we show that the problem of solving these classes of coupled nonlinear ODEs of any order, effectively reduces to solving a single first order nonlinear ODE. We then describe a procedure to derive explicit general solutions for the identified integrable coupled ODEs, when the above mentioned first order nonlinear ODE reduces to a Bernoulli equation. The equations which we generate and solve include the two coupled versions of modified Emden equations (in second order), coupled versions of Chazy equations (in third order) and their variants, higher dimensional coupled Ricatti and Abel chains as well as a new integrable chain and higher order equations.