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March  2011, 3(1): 1-22. doi: 10.3934/jgm.2011.3.1

Superposition rules and second-order Riccati equations

José F. Cariñena 1, and  Javier de Lucas Araujo 2,

1. 

Departamento de Física Teórica and IUMA, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50.009, Zaragoza, Spain
2. 

Institute of Mathematics, Polish Academy of Sciences, Śniadeckish 8, P.O. Box 21, 00-956, Warszawa, Poland

Received  June 2010 Revised  April 2011 Published  April 2011

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose various generalisations of this notion to second-order differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other second-order differential equations of mathematical and physical interest.
Keywords: Lie system, superposition rule., system of second-order differential equations, second-order Riccati equation, Quasi-Lie scheme.
Mathematics Subject Classification: 34A26, 34A34, 53Z0.
Citation: José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1
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show all references

References:
[1]

Numer. Algorithms, 15 (1997), 111-134. doi: 10.1023/A:1019262520178.  Google Scholar

[2]

J. Math. Phys., 31 (1990), 2528-2534. doi: 10.1063/1.528997.  Google Scholar

[3]

J. Math. Phys., 32 (1991), 1480-1490. doi: 10.1063/1.529306.  Google Scholar

[4]

J. Phys. A, 42 (2009), 335206. doi: 10.1088/1751-8113/42/33/335206.  Google Scholar

[5]

J. Phys A, 43 (2010), 305201. doi: 10.1088/1751-8113/43/30/305201.  Google Scholar

[6]

Rep. Math. Phys., 60 (2007), 237-258.  Google Scholar

[7]

J. Phys.: Conf. Ser., 175 (2009), 012009. doi: 10.1088/1742-6596/175/1/012009.  Google Scholar

[8]

J. Math. Phys., 50 (2009), 103515.  Google Scholar

[9]

Phys. Lett. A, 372 (2008), 5385-5389.  Google Scholar

[10]

J. Phys. A, 41 (2008), 304029. doi: 10.1088/1751-8113/41/30/304029.  Google Scholar

[11]

J. Theoret. Phys. (2010). Available from: http://www.springerlink.com/content/51mu8h705025m158/. Google Scholar

[12]

SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008), 031.  Google Scholar

[13]

Banach Center Publ., 59, Polish Acad. Sci., Warsaw, (2003), 143-162.  Google Scholar

[14]

J. Math. Phys., 46 (2005), 062703. doi: 10.1063/1.1920287.  Google Scholar

[15]

Phys. Rev. E, 72 (2005), 066203. doi: 10.1103/PhysRevE.72.066203.  Google Scholar

[16]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2451-2476.  Google Scholar

[17]

J. Nonlinear Math. Phys., 15 (2008), 365-382. doi: 10.2991/jnmp.2008.15.4.2.  Google Scholar

[18]

Dover Publications, New York, 1962.  Google Scholar

[19]

Phys. Rev. A, 41 (1990), 4166-4173.  Google Scholar

[20]

SIAM J. Math. Anal., 8 (1977), 1032-1037. doi: 10.1137/0508079.  Google Scholar

[21]

"Wave Phenomena: Modern Theory and Applications," North Holland Mathematics Studies, 97, Amsterdam, (1984), 199-210.  Google Scholar

[22]

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[24]

R. Flores-Espinoza, Periodic first integrals for Hamiltonian systems of Lie type,, \arXiv{1004.1132}., ().   Google Scholar

[25]

Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 485-494. doi: 10.3934/dcdsb.2008.10.485.  Google Scholar

[26]

Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950.  Google Scholar

[27]

J. Phys. A, 32 (1999), 3931-3937. doi: 10.1088/0305-4470/32/21/306.  Google Scholar

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[34]

Teubner, Leipzig, 1893.  Google Scholar

[35]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3005-3021.  Google Scholar

[36]

J. Math. Phys., 27 (1986), 14-23. doi: 10.1063/1.527381.  Google Scholar

[37]

Acta Math., 25 (1902), 1-85.  Google Scholar

[38]

J. Math. Phys., 50 (2009), 102701. doi: 10.1063/1.3204075.  Google Scholar

[39]

J. Math. Anal. Appl., 216 (1997), 246-264. doi: 10.1006/jmaa.1997.5674.  Google Scholar

[40]

J. Franklin Inst., 344 (2007), 391-398. Google Scholar

[41]

Ann. Sci. École Norm. Sup., 10 (1893), 53-64.  Google Scholar

[42]

Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), H1-H33.  Google Scholar

[43]

Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1895), F1-F26.  Google Scholar

[44]

C.R. Math. Acad. Sci. Paris, 137 (1903), 1033-1035. Google Scholar

[45]

Lecture Notes in Phys., 189 (1983), 263-305. doi: 10.1007/3-540-12730-5_12.  Google Scholar

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