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**Linearization: geometric, complex, and conditional.**
*(English)*
Zbl 1269.34045

Summary: Lie symmetry analysis provides a systematic method of obtaining exact solutions of nonlinear (systems of) differential equations, whether partial or ordinary. Of special interest is the procedure that Lie developed to transform scalar nonlinear second-order ordinary differential equations to linear form. Not much work was done in this direction to start with, but recently there have been various developments. Here, first the original work of Lie (and the early developments on it), and then more recent developments based on geometry and complex analysis, apart from Lie’s own method of algebra (namely, Lie group theory), are reviewed. It is relevant to mention that much of the work is not linearization but uses the base of linearization.

### MSC:

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34C14 | Symmetries, invariants of ordinary differential equations |

34A05 | Explicit solutions, first integrals of ordinary differential equations |

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\textit{A. Qadir}, J. Appl. Math. 2012, Article ID 303960, 30 p. (2012; Zbl 1269.34045)

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### References:

[1] | E. Galois, “Analyse d’un memoire sur la rsolution algbrique des quations,” Bulletin des Sciences Mathématiques Journal, vol. 13, p. 271, 1830. |

[2] | S. Lie, “Theorie der Transformationsgruppen,” Mathematische Annalen, vol. 16, p. 441, 1880. · JFM 12.0292.01 |

[3] | S. Lie, “Klassifikation und integration von gewönlichen differentialgleichungenzwischen x, y, die eine gruppe von transformationen gestaten,” Archiv for Matematik, vol. 8, no. 9, p. 187, 1883. · JFM 15.0751.03 |

[4] | G. Sheffers, Lectures on Differential Equations with Known Infinitesimal Transformations, Teubner, Leipzig, Germany, 1891. |

[5] | Differential Equations, Chelsea Publishing, New York, NY, USA, 1967. · Zbl 0221.34004 |

[6] | A. Tresse, “Sur les invariants differentiels des groupes continus de transformations,” Acta Mathematica, vol. 18, pp. 181-188, 1894. · JFM 25.0641.01 |

[7] | S. S. Chern, “Sur la geometrie d’une equation differentielle du troiseme orde,” Comptes Rendus de l’Académie des Sciences, vol. 204, pp. 1227-1229, 1937. · Zbl 0016.16401 |

[8] | S. S. Chern, “The geometry of the differential equation y\?=F(x,y,y\(^{\prime}\),y\(^{\prime\prime}\)),” Science Reports of National Tsing Hua University, vol. 4, pp. 97-111, 1940. · Zbl 0024.19801 |

[9] | G. Grebot, “The linearization of third-order ODEs,” submitted. · Zbl 0869.34007 |

[10] | G. Grebot, “The characterization of third order ordinary differential equations admitting a transitive fiber-preserving point symmetry group,” Journal of Mathematical Analysis and Applications, vol. 206, no. 2, pp. 364-388, 1997. · Zbl 0869.34007 |

[11] | F. M. Mahomed and P. G. L. Leach, “Symmetry Lie algebras of nth order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 151, no. 1, pp. 80-107, 1990. · Zbl 0719.34018 |

[12] | S. Neut and M. Petitot, “La geometrie de l’equation y\?=f(x,y,y\(^{\prime}\),y\(^{\prime\prime}\)),” Comptes Rendus de l’Académie des Sciences Series I, vol. 335, pp. 515-518, 2002. · Zbl 1016.34007 |

[13] | N. H. Ibragimov and S. V. Meleshko, “Linearization of third-order ordinary differential equations by point and contact transformations,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 266-289, 2005. · Zbl 1082.34003 |

[14] | N. H. Ibragimov, S. V. Meleshko, and S. Suksern, “Linearization of fourth-order ordinary differential equations by point transformations,” Journal of Physics A, vol. 41, no. 23, Article ID 235206, 19 pages, 2008. · Zbl 1151.34029 |

[15] | V. M. Gorringe and P. G. L. Leach, “Lie point symmetries for systems of 2nd order linear ordinary differential equations,” Quaestiones Mathematicae, vol. 11, pp. 95-117, 1988. · Zbl 0649.34018 |

[16] | C. Wafo Soh and F. M. Mahomed, “Symmetry breaking for a system of two linear second-order ordinary differential equations,” Nonlinear Dynamics, vol. 22, no. 1, pp. 121-133, 2000. · Zbl 0961.34023 |

[17] | C. W. Soh and F. M. Mahomed, “Linearization criteria for a system of second-order ordinary differential equations,” International Journal of Non-Linear Mechanics, vol. 36, no. 4, pp. 671-677, 2001. · Zbl 1345.34012 |

[18] | A. V. Aminova and N. A.-M. Aminov, “Projective geometry of systems of differential equations: general conceptions,” Tensor N S, vol. 62, pp. 65-86, 2000. · Zbl 1121.53304 |

[19] | A. V. Aminova and N. A.-M. Aminov, “Projective geometry of systems of second-order differential equations,” Sbornik, vol. 197, pp. 951-955, 2006. · Zbl 1143.53310 |

[20] | T. Feroze, F. M. Mahomed, and A. Qadir, “The connection between isometries and symmetries of geodesic equations of the underlying spaces,” Nonlinear Dynamics, vol. 45, no. 1-2, pp. 65-74, 2006. · Zbl 1100.53038 |

[21] | F. M. Mahomed and A. Qadir, “Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations,” Nonlinear Dynamics, vol. 48, no. 4, pp. 417-422, 2007. · Zbl 1184.34045 |

[22] | F. M. Mahomed and A. Qadir, “Invariant linearization criteria for systems of cubically nonlinear second-order ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 16, no. 3, pp. 283-298, 2009. · Zbl 1190.34045 |

[23] | Y. Y. Bagderina, “Linearization criteria for a system of two second-order ordinary differential equations,” Journal of Physics A, vol. 43, Article ID 465201, 14 pages, 2010. · Zbl 1213.34056 |

[24] | V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, “On the complete integrability and linearization of nonlinear ordinary differential equations. V. linearization of coupled second-order equations,” Proceedings of the Royal Society A, vol. 465, no. 2108, pp. 2369-2389, 2009. · Zbl 1186.34050 |

[25] | V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, “A systematic method of finding linearizing transformations for nonlinear ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 19, Article ID 1250012, 21 pages, 2012. · Zbl 1255.34038 |

[26] | S. Ali, Complex symmetry analysis [Ph.D. thesis], NUST CAMP, 2009. |

[27] | S. Ali, F. M. Mahomed, and A. Qadir, “Complex Lie symmetries for scalar second-order ordinary differential equations,” Nonlinear Analysis, vol. 10, no. 6, pp. 3335-3344, 2009. · Zbl 1187.34044 |

[28] | R. Penrose, The Road To Relaity: A Complete Guide To the Laws of the Universe, A. Knopf, New York, NY, USA, 2005. · Zbl 1188.00007 |

[29] | F. M. Mahomed, “Symmetry group classification of ordinary differential equations: survey of some results,” Mathematical Methods in the Applied Sciences, vol. 30, pp. 1995-2012, 2007. · Zbl 1135.34029 |

[30] | E. Laguerre, “Sur les equations differentielles lineaires du troisieme ordre,” Comptes Rendus, vol. 88, pp. 116-119, 1879. · JFM 11.0235.01 |

[31] | E. Laguerre, “Sur quelques invariants des equations differentielles,” Comptes Rendus, vol. 88, pp. 224-227, 1879. · JFM 11.0235.02 |

[32] | S. V. Meleshko, “On linearization of third-order ordinary differential equations,” Journal of Physics A, vol. 39, no. 49, pp. 15135-15145, 2006. · Zbl 1118.34034 |

[33] | T. Feroze and A. Qadir, “Another representation of the Lie algebra of second-order vector differential equations,” Differential Equations and Nonlinear Mechanics, vol. 2009, Article ID 152698, 3 pages, 2009. · Zbl 1220.34054 |

[34] | F. Gonzalez Gascon and A. Gonzalez-Lopez, “Symmetries of systems of differential equations IV,” Journal of Mathematical Physics, vol. 24, pp. 2006-2021, 1983. · Zbl 0564.35081 |

[35] | E. Fredericks, F. M. Mahomed, E. Momoniat, and A. Qadir, “Constructing a space from the geodesic equations,” Computer Physics Communications, vol. 179, no. 6, pp. 438-442, 2008. · Zbl 1197.53005 |

[36] | A. H. Bokhari and A. Qadir, “A prescription for n-dimensional vierbeins,” ZAMP Zeitschrift für angewandte Mathematik und Physik, vol. 36, no. 1, pp. 184-188, 1985. · Zbl 0579.53022 |

[37] | S. Ali, F. M. Mahomed, and A. Qadir, “Linearizability criteria for systems of two secondorder differential equations by complex methods,” Nonlinear Dynamics, vol. 66, pp. 77-88, 2011. · Zbl 1284.35018 |

[38] | M. Tsamparlis and A. Paliathanasis, “Lie and Noether symmetries of geodesic equations and collineations,” General Relativity and Gravitation, vol. 42, no. 12, pp. 2957-2980, 2010. · Zbl 1255.83031 |

[39] | C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Physical Review Letters, vol. 98, Article ID 040403, 4 pages, 2007. · Zbl 1228.81027 |

[40] | C. M. Bender, D. D. Holm, and D. W. Hook, “Complex trajectories of a simple Pendulum,” Journal of Physics A, vol. 40, pp. F81-F89, 2007. · Zbl 1112.70020 |

[41] | C. M. Bender, D. D. Holm, and D. W. Hook, “Complexified dynamical systems,” Journal of Physics A, vol. 40, pp. F793-F804, 2007. · Zbl 1120.37050 |

[42] | C. M. Bender, D. D. Holm, and D. W. Hook, “Quantum effects in classical systems having complex energy,” Journal of Physics A, vol. 41, Article ID 352003, 15 pages, 2008. · Zbl 1145.81416 |

[43] | C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Reports on Progress in Physics, vol. 70, no. 6, pp. 947-1018, 2007. · Zbl 1143.81312 |

[44] | C. M. Bender and H. F. Jones, “Interactions of Hermitian and non-Hermitian Hamiltonians,” Journal of Physics A, vol. 41, Article ID 244006, 8 pages, 2008. · Zbl 1140.81377 |

[45] | S. Ali, F. M. Mahomed, and A. Qadir, “Complex Lie symmetries for variational problems,” Journal of Nonlinear Mathematical Physics, vol. 15, pp. 124-133, 2008. · Zbl 1362.70024 |

[46] | M. Safdar, S. Ali, and F. M. Mahomed, “Linearization of systems of four second-order ordinary differential equations,” Pramana, vol. 77, pp. 581-594, 2011. |

[47] | M. Ayub, M. Khan, and F. M. Mahomed, “Algebraic linearization criteria for systems of second-order ordinary differential equations,” Nonlinear Dynamics, vol. 67, pp. 2053-2062, 2012. · Zbl 1251.34053 |

[48] | J. Merker, “Characterization of the newtonian free particle system in m \geq 2 dependent variables,” Acta Applicandae Mathematicae, vol. 92, no. 2, pp. 125-207, 2006. · Zbl 1330.34016 |

[49] | S. Ali, A. Qadir, and M. Safdar, “Inequivalence of classes of linearizable systems of cubically semi linear ordinary differential equations obtained by real and complex symmetry analysis,” Applied Mathematics and Computation, vol. 16, pp. 923-934, 2011. · Zbl 1252.34041 |

[50] | M. Safdar, S. Ali, and A. Qadir, “On the symmetry solutions of systems of two second-order ordinary differential equations not solvable by symmetry analysis,” submitted, http://arxiv.org/abs/1104.3837v2. · Zbl 1252.34041 |

[51] | F. M. Mahomed and A. Qadir, “Conditional linearizability criteria for third-order ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 15, pp. 25-35, 2008. · Zbl 1362.34062 |

[52] | F. M. Mahomed and A. Qadir, “Conditional linearizability of fourth-order semi-linear ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 16, no. 1, pp. 165-178, 2009. · Zbl 1362.34061 |

[53] | F. M. Mahomed, I. Naeem, and A. Qadir, “Conditional linearizability criteria for a system of third-order ordinary differential equations,” Nonlinear Analysis, vol. 10, no. 6, pp. 3404-3412, 2009. · Zbl 1181.34049 |

[54] | F. M. Mahomed and A. Qadir, “Classification of ordinary differential equations by conditional linearizability and symmetry,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, pp. 573-584, 2012. · Zbl 1246.34033 |

[55] | H. M. Dutt and A. Qadir, “Meleshko’s method of “linearization” for fourth order, scalar, ordinary differential equations,” Work in progress. |

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