Theses and Dissertations

Date of Award


Document Type


Degree Name

Master of Science (MS)



First Advisor

Dr. Zhaosheng Feng

Second Advisor

Dr. Tim Huber

Third Advisor

Dr. Zhijun Qiao


In this paper, we apply the theory of Lie symmetry to study a generalized second-order nonlinear differential equation, which includes several physical nonlinear oscillators such as force-free Helmholtz oscillator, force-free Duffing and Duffing-van der Pol oscillators, modified Emden-type equation and its hierarchy etc, and investigate the dynamical properties of this rather general equation. We identify and classify several new integrable cases for arbitrary values of exponents, which determine the tangent vector as well as the infinitesimal generator. Using the Lie point symmetry, we find the useful infinitesimal generators and canonical coordinates, and obtain the first integrals of the second-order nonlinear systems under certain parametric conditions. It shows that many classical integrable nonlinear oscillators can be derived as subcases of the obtained results and significantly enlarge the list of integrable equations that exists in the contemporary literature. Applications to wave equations of Korteweg-de Vries- Burgers-type equations are also presented.


Copyright 2016 Xiaoyan Li. All Rights Reserved.

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