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A group theoretical identification of integrable cases of the Liénard-type equation . I. Equations having nonmaximal number of Lie point symmetries

### Abstract

We carry out a detailed Lie point symmetry group classification of the Liénard-type equation, , where and are arbitrary smooth functions of . We divide our analysis into two parts. In the present first part we isolate equations that admit lesser parameter Lie point symmetries, namely, one, two, and three parameter symmetries, and in the second part we identify equations that admit maximal (eight) parameter Lie point symmetries. In the former case the invariant equations form a family of integrable equations, and in the latter case they form a class of linearizable equations (under point transformations). Further, we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent Hamiltonians. Several of these equations are being identified for the first time from the group theoretical analysis.

© 2009 American Institute of Physics

Received 11 February 2009
Accepted 02 July 2009
Published online 11 August 2009
Publisher error corrected 14 August 2009

Acknowledgments:
One of us (S.N.P.) is grateful to the Centre for Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli, for warm hospitality. The work of S.N.P. forms part of a Department of Science and Technology, Government of India sponsored research project. The work of M.S. forms part of a research project sponsored by National Board for Higher Mathematics, Government of India. The work of M.L. forms part of a Department of Science and Technology (DST), Ramanna Fellowship, and is also supported by a DST-IRHPA research project.

Article outline:

I. INTRODUCTION
II. DETERMINING EQUATIONS FOR THE INFINITESIMAL SYMMETRIES
A. Alternate way
III. LIE SYMMETRIES OF LIÉNARD-TYPE SYSTEMS–LESSER PARAMETER SYMMETRIES: CASE
A. Two-parameter Lie point symmetries
1. Case 1
2. Integrability of Eq. (32) for arbitrary values of
3. Integrable equations with
4. Integrable equations with
5. Case *2 * : Integrable equation
B. Nonexistence of three-parameter symmetry group in the general case
IV. ETs
V. LIE SYMMETRIES OF Eq. (1) WITH OR
VI. CONCLUSIONS

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2009-08-11

2016-02-10

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