A simple and unified approach to identify integrable nonlinear oscillators and systems

Abstract

In this paper, we consider a generalized second order nonlinear ordinary differential equation of the formẍ + (k 1 x q + k 2 )ẋ + k 3 x 2q+1 + k 4 x q+1 + λ 1 x = 0, where k i 's, i = 1, 2, 3, 4, λ 1 and q are arbitrary parameters, which includes several physically important nonlinear oscillators such as the simple harmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator, force-free Duffing and Duffing-van der Pol oscillators, modified Emden type equation and its hierarchy, generalized Duffing-van der Pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation. We identify several new integrable cases for arbitrary value of the exponent q, q ∈ R. The q = 1 and q = 2 cases are analyzed in detail and the results are generalized to arbitrary q. Our results show that many classical integrable nonlinear oscillators can be derived as sub-cases of our results and significantly enlarge the list of integrable equations that exist in the contemporary literature. To explore the above underlying results we use the recently introduced generalized extended Prelle-Singer procedure applicable to second order ODEs. As an added advantage of the method we not only identify integrable regimes but also construct integrating factors, integrals of motion and general solutions for the integrable cases, wherever possible, and bring out the mathematical structures associated with each of the integrable cases.

Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries

“…In [10,11] Duarte et al have extended the PS method to include such situations. Subsequently in a series of papers [6,8,9] Chandrasekar et al have uncovered rational and even non rational first integrals for a large class of oscillator type equations, by appropriately modifying and also extending the basic idea behind the PS procedure.…”

“…In [9], the authors have studied this system for q = arbitrary and deduced a number of new completely integrable cases.…”

“…Let us briefly review the method used by the authors to deduce first integrals of oscillator type systems under very general conditions [7,8,9]. According to these calculations the equation of motion for the second-order ODE is written in the form:…”

“…Next we make the ansatz 9) and after inserting it into (5.8), obtain the following equation for determining T (x, v),…”

“…Most recently Lakshmanan and his coworkers have generalized and used the extended Prelle-Singer method to obtain the first integrals and general solutions for a class of nonlinear equations [6,7,9]. They have also devised a procedure to construct a transformation which removes the time-dependent part from the first integral and provides the general solution by quadrature [8].…”