Painleve analysis, Lie symmetries, and integrability of coupled nonlinear oscillators of polynomial type

Abstract

In recent investigations on nonlinear dynamics, the singularity structure analysis pioneered by Kovalevskaya, Painleve and contempories, which stresses the meromorphic nature of the solutions of the equations of motion in the complex-time plane, is found to play an increasingly important role. Particularly, soliton equations have been found to be associated with the so-called Painleve property, which implies that the solutions are free from movable critical points/manifolds. Finite-dimensional integrable dynamical systems have also been found to possess such a property. In this review, after briefly presenting the historical developments and various features of the Painleve (P) method, we demonstrate how it provides an effective tool in the analysis of nonlinear dynamical systems, starting from simple examples. We apply this method to several important coupled nonlinear oscillators governed by generic Hamiltonians of polynomial type with two, three and arbitrary (N) degrees of freedom and classify all the P-cases. Sufficient numbers of involutive integrals of motion for each of the P-cases are constructed by employing other direct methods. In particular, we examine the question of integrability from the viewpoint of symmetries, explicitly demonstrate the existence of nontrivial extended Lie symmetries for the P-cases, and obtain the required integrals of motion by direct integration of symmetries. Furthermore, we briefly explain how the singularity structure analysis can be used to understand some of the intrinsic properties of nonintegrability and chaos with special reference to the two-coupled quartic anharmonic oscillators and Henon-Heiles systems.