Linearizabiliy and Lax representations for cubic autonomous and non-autonomous nonlinear oscillators

Abstract

In this work we consider a family of cubic, with respect to the first derivative, second-order ordinary differential equations. We study linearizability conditions for this family of equations via generalized nonlocal transformations. We construct linearizability criteria in the explicit form for some particular cases of these transformations. We show that each linearizable equation admits a quadratic rational first integral. Moreover, we demonstrate that generalized nonlocal transformations under certain conditions preserve Lax integrability for equations from the considered family. Consequently, we find that any linearizable equation from the considered family possesses a Lax representation. This provides a connection between two different approaches for studying integrability and demonstrates that the Lax technique can be effectively applied to finding and classifying dissipative integrable autonomous and non-autonomous nonlinear oscillators. We illustrate our results by several examples of linearizable equations, including a non-autonomous generalization of the Rayleigh–Duffing oscillator, a family of chemical oscillators and a generalized Duffing–Van der Pol oscillator.