PAINLEVE ANALYSIS, LIE SYMMETRIES, AND INTEGRABILITY OF COUPLED NONLINEAR OSCILLATORS OF POLYNOMIAL TYPE

In recent investigations on nonlinear dynamics, the singularity struct ure analysis pioneered by Kovalevskaya, Painleve and contempories, whi ch stresses the meromorphic nature of the solutions of the equations o f motion in the complex-time plane, is found to play an increasingly i mportant role. Particularly, soliton equations have been found to be a ssociated with the so-called Painleve property, which implies that the solutions are free from movable critical points/manifolds. Finite-dim ensional integrable dynamical systems have also been found to possess such a property. In this review, after briefly presenting the historic al developments and various features of the Painleve (P) method, we de monstrate how it provides an effective tool in the analysis of nonline ar dynamical systems, starting from simple examples. We apply this met hod to several important coupled nonlinear oscillators governed by gen eric Hamiltonians of polynomial type with two, three and arbitrary (N) degrees of freedom and classify all the P-cases. Sufficient numbers o f involutive integrals of motion for each of the P-cases are construct ed by employing other direct methods. In particular, we examine the qu estion of integrability from the viewpoint of symmetries, explicitly d emonstrate the existence of nontrivial extended Lie symmetries for the P-cases, and obtain the required integrals of motion by direct integr ation of symmetries. Furthermore, we briefly explain how the singulari ty 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.