A Simple and Unified Approach to Identify Integrable Nonlinear Oscillators and Systems

摘要

In this paper, we consider a generalized second order nonlinear ordinarydifferential equation of the form$\ddot{x}+(k_1x^q+k_2)\dot{x}+k_3x^{2q+1}+k_4x^{q+1}+\lambda_1x=0$, where$k_i$'s, $i=1,2,3,4$, $\lambda_1$ and $q$ are arbitrary parameters, whichincludes several physically important nonlinear oscillators such as the simpleharmonic oscillator, anharmonic oscillator, force-free Helmholtz oscillator,force-free Duffing and Duffing-van der Pol oscillators, modified Emden typeequation and its hierarchy, generalized Duffing-van der Pol oscillator equationhierarchy and so on and investigate the integrability properties of this rathergeneral equation. We identify several new integrable cases for arbitrary valueof the exponent $q, q\in R$. The $q=1$ and $q=2$ cases are analyzed in detailand the results are generalized to arbitrary $q$. Our results show that manyclassical integrable nonlinear oscillators can be derived as sub-cases of ourresults and significantly enlarge the list of integrable equations that existin the contemporary literature. To explore the above underlying results we usethe recently introduced generalized extended Prelle-Singer procedure applicableto second order ODEs. As an added advantage of the method we not only identifyintegrable regimes but also construct integrating factors, integrals of motionand general solutions for the integrable cases, wherever possible, and bringout the mathematical structures associated with each of the integrable cases.