Lie Algebra Classification, Conservation Laws, and Invariant Solutions for a Generalization of the Levinson–Smith Equation

Abstract

We obtain the optimal system’s generating operators associated with a generalized Levinson–Smith equation; this one is related to the Liénard equation which is important for physical, mathematical, and engineering points of view. The underlying equation has applications in mechanics and nonlinear dynamics as well. This equation has been widely studied in the qualitative scheme. Here, we treat the equation by using the Lie group method, and we obtain certain operators; using those operators, we characterized all invariants solutions associated with the generalized equation of Levinson Smith considered in this paper. Finally, we classify the Lie algebra associated with the given equation.

Lie group symmetries' complete classification for a generalized Chazy equation and its equivalence group

“…Ideas to develop in future works could be calculating the conservation laws for (4), using the symmetries presented in Proposition 1, also it is possible to investigate the contact symmetries as was presented in [24,14,6] and try to obtain for the symmetries associated with the equation ( 4), the classification of its Lie algebra as was presented in [21].…”

Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

“…Recently, the Lie group method approach has been applied to solve and analyze different problems in many scientific fields, e.g., in [25], the authors applied the Lie symmetry method to investigate some solutions for pZK equation, which models the nonlinear propagation of dust-ion acoustic solitary waves and shocks, and in [11] the authors studied the invariance of stochastic differential equations under random diffeomorphisms and established the determining equations for random Lie point symmetries of stochastic differential equations. Some references in the latest progress using Lie group symmetries can be found in [1,10,13,27,37,40] and references therein, some recent publications in the area of Lie symmetry groups, optimal system, and invariant solutions can be found in [2,21,22,23,24,30,31,38].…”

Lie algebra classification for the Chazy equation and further topics related with this algebra

“…For example in [8] it was presented that the Lie algebra associated to generalizations of the Kummer-Schwarz equation in a particular case is ⊕ . Moreover, in [9] the authors classify the Lie algebra corresponding for a Generalization of the Levinson-Smith equation, and it is proved that this Lie algebra is a solvable nonnilpotent five dimensional Lie algebra which is isomorphic to g5,35 in the MubarAkzyanov's classification.…”