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

• Autores: Danilo García Hernández, Oscar Mario Londoño Duque, Yeisson Acevedo, Gabriel Loaiza
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 39, Nº. 2, 2021, págs. 257-274
• Idioma: español
• DOI: 10.18273/revint.v39n2-2021007
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    Obtenemos la clasificación completa del grupo de simetría de Lie y los operadores generadores del sistema optimal asociados a un caso particular de la ecuación de Kummer - Schwarz generalizada. Utilizando esos operadores, caracterizamos todas las soluciones invariantes, se encontraron soluciones alternativas para la ecuación estudiada y se clasifica el álgebra de Lie asociada al grupo de simetría.

  • English

    We obtain the complete classification of the Lie symmetry groupand the optimal system’s generating operators associated with a particularcase of the generalized Kummer - Schwarz equation. Using those operatorswe characterize all invariant solutions, alternative solutions were found for theequation studied and the Lie algebra associated with the symmetry group isclassified.

• Referencias bibliográficas
  • Referencias Ali M.R. and Sadat R., “Lie symmetry analysis, new group invariant for the (3 + 1) - dimensional and variable coefficients...
  • Alimirzaluo E., Nadjafikhah M. and Manafian J., “Some new exact solutions of (3 + 1) - dimensional burgers system via lie symmetry analysis”,...
  • Bozhkov Y.D. and Ramos P., “On the generalizations of the Kummer-Schwarz equation”, Nonlinear Anal. Optim., 192 (2020), 111–691. doi: 10.1016/j.na.2019.111691.
  • Bluman G. and Kumei S., Symmetries and Differential Equations, Springer Science & Business Media, vol. 81, New York, 1989.
  • Bluman G. and Anco S., Symmetry and integration methods for differential equations, Springer Science & Business Media, vol. 154, New York,...
  • Bluman G., Cheviakov A. and Anco S., Applications of symmetry methods to partial differential equations, Springer, vol. 168, New York, 2010.
  • Cariñena J.F. and De Lucas J., “Applications of lie systems in dissipative MilnePinney equations”, Int. J. Geom. Methods Mod. Phys., 6 (2009),...
  • Cantwell B.J., Introduction to Symmetry Analysis, Cambridge University Press, Cambridge, 2002.
  • Gainetdinova A.A., Ibragimov N.H. and Meleshko S.V., “Group classification of ODE y ′′′ = F(x, y, y′ )′′, Commun. Nonlinear Sci. Numer....
  • Ghose-Choudhury A. et al., “Noetherian symmetries of noncentral forces with drag term”, Int. J. Geom. Methods Mod. Phys., 14 (2017), No. 2,...
  • Gaeta G. and Spadaro F., Random Lie - point symmetries of stochastic differential equations, AIP Publishing LLC, 5th ed., vol. 58, 2017.
  • Gibbons G.W., “Dark energy and the Schwarzian derivative”, arXiv:1403.5431.
  • Hu W., et al., “Symmetry breaking of infinite-dimensional dynamic system”, Appl. Math. Lett., 103 (2020), 106–207. doi: 10.1016/j.aml.2019.106207.
  • Hydon P.E., “Discrete point symmetries of ordinary differential equations”, Proc. R. Soc. Lond. Ser., 454 (1998), No. 1975, 1961–1972. doi:...
  • Hydon P.E. and Crighton D., Symmetry methods for differential equations: a beginner’s guide, Cambridge University Press, vol. 22, Cambridge,...
  • Humphreys J.E., Introduction to Lie algebras and representation theory, Springer-Verlag, 1st ed., vol. 9, New York, 2012.
  • Hussain Z., Sulaiman M. and Sackey E., “Optimal system of subalgebras and invariant solutions for the Black-Scholes equation”, Thesis (MSc),...
  • Ibragimov N.H. and Nucci M.C., “Integration of Third Order Ordinary Differential Equations by Lie’s Method: Equations Admitting Three - Dimensional...
  • Ibragimov N.H., CRC Handbook of Lie Group Analysis of Differential Equations, CRC Press, vol. 3, 1995.
  • Khudija B., “Particular solutions of ordinary differential equations using discrete symmetry groups”, Symmetry., 12 (2020), No. 1, 180. doi:...
  • Kumar S., Ma W. X. and Kumar A., “Lie symmetries, optimal system and group - invariant solutions of the (3+1) - dimensional generalized...
  • Kumar S., Kumar D. and Kumar A., “Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons...
  • Kumar S., Kumar D. and Wazwaz A.M., “Lie symmetries, optimal system, group invariant solutions and dynamical behaviors of solitary wave solutions...
  • Kumar S., Almusawa H. and Kumar A., “Some more closed-form invariant solutions and dynamical behavior of multiple solitons for the (2 +...
  • Kumar D. and Kumar S., “Solitary wave solutions of pZK equation using Lie point symmetries” Eur. Phys. J. Plus., 135 (2020), No. 2, 1–19....
  • Leach P.G, “Symmetry and singularity properties of the generalised KummerSchwarz and related equations”, J. Math. Anal., 348 (2008), No. 1,...
  • Leach P.G. and Paliathanasis A., “Symmetry analysis for a fourth-order noisereduction partial differential equation”, Quaest. Math., (2020),...
  • Lie S., “Theorie der transformationsgruppen I”, Mathematische Annalen., 16 (1880), No. 4, 441–528. doi: 10.1007/BF01446218.
  • Llibre J. and Vidal C., “Global dynamics of the Kummer-Schwarz differential equation”, Mediterr. J. Math., 11 (2014), No. 2, 477-486. doi:...
  • Loaiza G., Acevedo Y., Duque O.M.L. and García D., “Lie algebra classification, conservation laws, and invariant solutions for a generalization...
  • Lu H. and Zhang Y., “Lie symmetry analysis, exact solutions, conservation laws and bäcklund transformations of the gibbons-tsarev equation”,...
  • Mertens T.G., Turiaci G.J. and Verlinde H.L., “Solving the Schwarzian via the conformal bootstrap”, Journal High Energy Phys., 2017 (2017),...
  • Noether E., Invariante Variationsprobleme, Mathematisch-physikalische Klasse, 2nd ed., 1918.
  • Ovsienko V. and Tabachnikov S., “What is the Schwarzian derivative”, Notices of the AMS., 56 (2009), No. 1, 34-36.
  • Olver P.J., Applications of Lie Groups to Differential Equations, Springer-Verlag, 1st ed., vol. 107, New York, 1986.
  • Ovsyannikov L., Group analysis of differential equations, Academic Press, 1st ed., New York, 1982.
  • Paliathanasis A. and Leach P.G.L., “Symmetries and singularities of the Szekeres system”, Modern Phys. Lett. A., 381 (2017), No. 15, 1277–1280....
  • Paliathanasis A., “Lie symmetry analysis and one-dimensional optimal system for the generalized 2+1 kadomtsev - petviashvili equation”,...
  • Stephani H., Differential equations: Their solution using symmetries, Cambridge University Press, 1st ed., Cambridge, 1989.
  • Tian S.F., “Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized boussinesq water...
  • Zaitsev V.F. and Polyanin A.D., Handbook of exact solutions for ordinary differential equations, Chapman and Hall/CRC, 2nd ed., New York,...
  • Zewdie G., “Lie simmetries of junction conditions for radianting stars”, Thesis (MSc), University of KwaZulu - Natal, Durban, 2011, 77 p.