A Group Theoretical Identification of Integrable Cases of the Liénard Type Equation $ddot{x} fxdot{x} gx = 0$ : Part I: Equations having Non-maximal Number of Lie point Symmetries - Nonlinear Sciences > Exactly Solvable and InReport as inadecuate




A Group Theoretical Identification of Integrable Cases of the Liénard Type Equation $ddot{x} fxdot{x} gx = 0$ : Part I: Equations having Non-maximal Number of Lie point Symmetries - Nonlinear Sciences > Exactly Solvable and In - Download this document for free, or read online. Document in PDF available to download.

Abstract: We carry out a detailed Lie point symmetry group classification of theLi\-enard type equation, $\ddot{x}+fx\dot{x}+gx = 0$, where $fx$ and$gx$ are arbitrary smooth functions of $x$. We divide our analysis into twoparts. In the present first part we isolate equations that admit lesserparameter Lie point symmetries, namely, one, two and three parametersymmetries, and in the second part we identify equations that admit maximaleight parameter Lie-point symmetries. In the former case the invariantequations form a family of integrable equations and in the latter case theyform a class of linearizable equations under point transformations. Further,we prove the integrability of all of the equations obtained in the presentpaper through equivalence transformations either by providing the generalsolution or by constructing time independent Hamiltonians. Several of theseequations are being identified for the first time from the group theoreticalanalysis.



Author: S. N. Pandey, P. S. Bindu, M. Senthilvelan, M. Lakshmanan

Source: https://arxiv.org/



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