Chandrasekar, V. K. ; Pandey, S. N. ; Senthilvelan, M. ; Lakshmanan, M. (2006) A simple and unified approach to identify integrable nonlinear oscillators and systems Journal of Mathematical Physics, 47 (2). 023508_1-023508_37. ISSN 0022-2488
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Official URL: http://link.aip.org/link/?JMAPAQ/47/023508/1
Related URL: http://dx.doi.org/10.1063/1.2171520
Abstract
In this paper, we consider a generalized second-order nonlinear ordinary differential equation (ODE) of the form x..+(k1xq+k2)x.+k3x2q+1+k4 xq+1+λ1x = 0, where ki's, i = 1,2,3,4, λ1, 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 subcases of our results and significantly enlarge the list of integrable equations that exists 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.
Item Type: | Article |
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Source: | Copyright of this article belongs to American Institute of Physics. |
ID Code: | 19413 |
Deposited On: | 22 Nov 2010 12:38 |
Last Modified: | 17 May 2016 03:58 |
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