Lax pair and first integrals for two of nonlinear coupled oscillators

TL;DR

This paper derives a Lax pair and first integrals for a system of two nonlinear coupled oscillators, including special cases and reduction to a fourth-order ODE, aiding in understanding their integrability.

Contribution

The paper presents the first known Lax pair for the coupled oscillators system, enabling the derivation of first integrals and analysis of integrability.

Findings

Lax pair for the coupled oscillators system is established.

Two first integrals are obtained from the Lax pair.

System reduction to a fourth-order ODE is demonstrated.

Abstract

The system of two nonlinear coupled oscillators is studied. As partial case this system of equation is reduced to the Duffing oscillator which has many applications for describing physical processes. It is well known that the inverse scattering transform is one of the most powerful methods for solving the Cauchy problems of partial differential equations. To solve the Cauchy problem for nonlinear differential equations we can use the Lax pair corresponding to this equation. The Lax pair for ordinary differential or systems or for system ordinary differential equations allows us to find the first integrals, which also allow us to solve the question of integrability for differential equations. In this report we present the Lax pair for the system of coupled oscillators. Using the Lax pair we get two first integrals for the system of equations. The considered system of equations can be also reduced to the fourth-order ordinary differential equation and the Lax pair can be used for the ordinary differential equation of fourth order. Some special cases of the system of equations are considered.