A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators

Abstract

We explore a nonlocal connection between certain linear and nonlinear ordinary differential equations (ODEs), representing physically important oscillator systems, and identify a class of integrable nonlinear ODEs of any order. We also devise a method to derive explicit general solutions of the nonlinear ODEs. Interestingly, many well known integrable models can be accommodated into our scheme and our procedure thereby provides further understanding of these models.

12 pages. J. Phys. A: Math. Gen. 39 (2006) in press

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• A Group Theoretical Identification of Integrable Cases of the Liénard Type Equation $\ddot{x}+f(x)\dot{x}+g(x)=0$ : Part I: Equations having Non-maximal Number of Lie point Symmetries
• Nonlocal symmetries of Riccati and Abel chains and their similarity reductions
• Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study
• Exact solutions of the Liénard and generalized Liénard type ordinary non-linear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator
• Noether symmetries and the quantization of a Lienard-type nonlinear oscillator
• Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order
• On the integrability of the Abel and of the extended Liénard equations
• Recursive Generation of Isochronous Hamiltonian Systems
• Dynamics of the Modified Emden and pseudo-Modified Emden Equations: position-dependent mass, invariance and exact solvability
• A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case
• Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden type equations
• Quasi-harmonic spectra from branched Hamiltonians