# American Institute of Mathematical Sciences

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

## Superposition rules and second-order Riccati equations

 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.
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
##### References:
 [1] C. Arnold, Formal continued fractions solutions of the generalized second order Riccati equations, applications,, Numer. Algorithms, 15 (1997), 111. doi: 10.1023/A:1019262520178. Google Scholar [2] J. Beckers, L. Gagnon, V. Hussin and P. Winternitz, Superposition formulas for nonlinear superequations,, J. Math. Phys., 31 (1990), 2528. doi: 10.1063/1.528997. Google Scholar [3] S. E. Bouquet, M. R. Feix and P. G. L. Leach, Properties of second-order ordinary differential equations invariant under time translation and self-similar transformation,, J. Math. Phys., 32 (1991), 1480. doi: 10.1063/1.529306. Google Scholar [4] J. F. Cariñena, J. Grabowski and J. de Lucas, Quasi-Lie schemes: theory and applications,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/33/335206. Google Scholar [5] J. F. Cariñena, J. Grabowski and J. de Lucas, Lie families: theory and applications,, J. Phys A, 43 (2010). doi: 10.1088/1751-8113/43/30/305201. Google Scholar [6] J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations,, Rep. Math. Phys., 60 (2007), 237. Google Scholar [7] J. F. Cariñena, P. Guha and M. F. Rañada, A geometric approach to higher-order Riccati chain: Darboux polynomials and constants of the motion,, J. Phys.: Conf. Ser., 175 (2009). doi: 10.1088/1742-6596/175/1/012009. Google Scholar [8] J. F. Cariñena, P. G. L. Leach and J. de Lucas, Quasi-Lie systems and Emden-Fowler equations,, J. Math. Phys., 50 (2009). Google Scholar [9] J. F. Cariñena and J. de Lucas, A nonlinear superposition rule for solutions of the Milne-Pinney equation,, Phys. Lett. A, 372 (2008), 5385. Google Scholar [10] J. F. Cariñena, J. de Lucas and M. F. Rañada, Integrability of Lie systems and some of its applications in physics,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/30/304029. Google Scholar [11] J. F. Cariñena, J. de Lucas and M. F. Rañada, A geometric approach to integrability of Abel differential equations,, J. Theoret. Phys. (2010). Available from: , (2010). Google Scholar [12] J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008). Google Scholar [13] J. F. Cariñena and A. Ramos, Applications of Lie systems in quantum mechanics and control theory, in: "Classical and Quantum Integrability,'', Banach Center Publ., (2003), 143. Google Scholar [14] J. F. Cariñena, M. F. Rañada and M. Santander, Lagrangian formalism for nonlinear second-order Riccati systems: one-dimensional integrability and two-dimensional superintegrability,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1920287. Google Scholar [15] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Unusual Liénard-type nonlinear oscillator,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.066203. Google Scholar [16] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2451. Google Scholar [17] A. G. Choudhury, P. Guha and B. Khanra, Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries,, J. Nonlinear Math. Phys., 15 (2008), 365. doi: 10.2991/jnmp.2008.15.4.2. Google Scholar [18] H. T. Davis., "Introduction to Nonlinear Differential and Integral Equations,'', Dover Publications, (1962). Google Scholar [19] J. M. Dixon and J. A. Tuszyński, Solutions of a generalized Emden equation and their physical significance,, Phys. Rev. A, 41 (1990), 4166. Google Scholar [20] L. Erbe, Comparison theorems for second order Riccati equations with applications,, SIAM J. Math. Anal., 8 (1977), 1032. doi: 10.1137/0508079. Google Scholar [21] V. J. Ervin, W. F. Ames and E. Adams, Nonlinear waves in the pellet fusion process, in:, Wave Phenomena: Modern Theory and Applications, (1984), 199. Google Scholar [22] M. Euler, N. Euler and P. G. L. Leach, The Riccati and Ermakov-Pinney hierarchies,, J. Nonlinear Math. Phys., 14 (2007), 290. doi: 10.2991/jnmp.2007.14.2.10. Google Scholar [23] W. Fair and Y. L. Luke, Rational approximations to the solution of the second order Riccati equation,, Math. Comp., 20 (1966), 602. doi: 10.1090/S0025-5718-1966-0203906-X. Google Scholar [24] R. Flores-Espinoza, Periodic first integrals for Hamiltonian systems of Lie type,, \arXiv{1004.1132}., (). Google Scholar [25] I. A. García, J. Giné and J. Llibre, Liénard and Riccati differential equations related via Lie algebras,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 485. doi: 10.3934/dcdsb.2008.10.485. Google Scholar [26] V. V. Golubev, "Lectures on the Analytical Theory of Differential Equations,'', Gosudarstv. Izdat. Tehn.-Teor. Lit., (1950). Google Scholar [27] A. M. Grundland and D. Levi, On higher-order Riccati equations as Bäcklund transformations,, J. Phys. A, 32 (1999), 3931. doi: 10.1088/0305-4470/32/21/306. Google Scholar [28] A. Guldberg, Sur les équations différentielles ordinaires qui possèdent un système fondamental d'intégrales, (French) [On the differential equations admitting a fundamental system of integrals],, C.R. Math. Acad. Sci. Paris, 116 (1893), 964. Google Scholar [29] N. H. Ibragimov, "Elementary Lie Group Analysis and Ordinary Differential Equations,'', J. Wiley & Sons, (1999). Google Scholar [30] E. L. Ince, "Ordinary Differential Equations,'', Dover Publications, (1944). Google Scholar [31] A. Karasu and P. G. L. Leach, Nonlocal symmetries and integrable ordinary differential equations: $\ddot x + 3x\dot x + x^3 = 0$ and its generalizations,, J. Math. Phys., 50 (2009). Google Scholar [32] S. Lafortune and P. Winternitz, Superposition formulas for pseudounitary matrix Riccati equations,, J. Math. Phys., 37 (1996), 1539. doi: 10.1063/1.531448. Google Scholar [33] J. A. Lázaro-Camí and J. P. Ortega, Superposition rules and stochastic Lie-Scheffers systems,, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 910. Google Scholar [34] S. Lie and G. Scheffers, "Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen,'' (German) [Lectures on continuous groups with geometric (and other) applications],, Teubner, (1893). Google Scholar [35] A. B. Olde Daalhuis, Hyperasymptotics for nonlinear ODEs (II). The first Painlevé equation and a second-order Riccati equation,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3005. Google Scholar [36] M. A. del Olmo, M. A. Rodríguez and P. Winternitz, Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles,, J. Math. Phys., 27 (1986), 14. doi: 10.1063/1.527381. Google Scholar [37] P. Painlevé, Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, (French) [On second- and higher-order differential equations whose general integral is uniform],, Acta Math., 25 (1902), 1. Google Scholar [38] S. N. Pandey, P. S. Bindu, M. Senthilvelan and M. Lakshmanan, A group theoretical identification of integrable equations in the Liénard-type equation $\ddot x+f(x)+g(x)=0$. II. Equations having maximal Lie point symmetries,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3204075. Google Scholar [39] C. Rogers, W. K. Schief and P. Winternitz, Lie-theoretical generalization and discretization of the Pinney equation,, J. Math. Anal. Appl., 216 (1997), 246. doi: 10.1006/jmaa.1997.5674. Google Scholar [40] C. Tunç and E. Tunç, On the asymptotic behaviour of solutions of certain second-order differential equations,, J. Franklin Inst., 344 (2007), 391. Google Scholar [41] M. E. Vessiot, Sur une classe d'équations différentielles, (French) [On a class of differential equations],, Ann. Sci. École Norm. Sup., 10 (1893), 53. Google Scholar [42] M. E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, (French) [On the systems of first-order differential equations admitting a fundamental system of integrals],, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894). Google Scholar [43] M. E. Vessiot, Sur quelques équations différentielles ordinaires du second ordre, (French) [On certain second-order differential equations],, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1895). Google Scholar [44] G. Wallenberg, Sur l'équation différentielle de Riccati du second ordre, (French) [On the second-order Riccati equations],, C.R. Math. Acad. Sci. Paris, 137 (1903), 1033. Google Scholar [45] P. Winternitz, Lie groups and solutions of nonlinear differential equations,, Lecture Notes in Phys., 189 (1983), 263. doi: 10.1007/3-540-12730-5_12. Google Scholar

show all references

##### References:
 [1] C. Arnold, Formal continued fractions solutions of the generalized second order Riccati equations, applications,, Numer. Algorithms, 15 (1997), 111. doi: 10.1023/A:1019262520178. Google Scholar [2] J. Beckers, L. Gagnon, V. Hussin and P. Winternitz, Superposition formulas for nonlinear superequations,, J. Math. Phys., 31 (1990), 2528. doi: 10.1063/1.528997. Google Scholar [3] S. E. Bouquet, M. R. Feix and P. G. L. Leach, Properties of second-order ordinary differential equations invariant under time translation and self-similar transformation,, J. Math. Phys., 32 (1991), 1480. doi: 10.1063/1.529306. Google Scholar [4] J. F. Cariñena, J. Grabowski and J. de Lucas, Quasi-Lie schemes: theory and applications,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/33/335206. Google Scholar [5] J. F. Cariñena, J. Grabowski and J. de Lucas, Lie families: theory and applications,, J. Phys A, 43 (2010). doi: 10.1088/1751-8113/43/30/305201. Google Scholar [6] J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations,, Rep. Math. Phys., 60 (2007), 237. Google Scholar [7] J. F. Cariñena, P. Guha and M. F. Rañada, A geometric approach to higher-order Riccati chain: Darboux polynomials and constants of the motion,, J. Phys.: Conf. Ser., 175 (2009). doi: 10.1088/1742-6596/175/1/012009. Google Scholar [8] J. F. Cariñena, P. G. L. Leach and J. de Lucas, Quasi-Lie systems and Emden-Fowler equations,, J. Math. Phys., 50 (2009). Google Scholar [9] J. F. Cariñena and J. de Lucas, A nonlinear superposition rule for solutions of the Milne-Pinney equation,, Phys. Lett. A, 372 (2008), 5385. Google Scholar [10] J. F. Cariñena, J. de Lucas and M. F. Rañada, Integrability of Lie systems and some of its applications in physics,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/30/304029. Google Scholar [11] J. F. Cariñena, J. de Lucas and M. F. Rañada, A geometric approach to integrability of Abel differential equations,, J. Theoret. Phys. (2010). Available from: , (2010). Google Scholar [12] J. F. Cariñena, J. de Lucas and M. F. Rañada, Recent applications of the theory of Lie systems in Ermakov systems,, SIGMA Symmetry Integrability Geom. Methods Appl., 4 (2008). Google Scholar [13] J. F. Cariñena and A. Ramos, Applications of Lie systems in quantum mechanics and control theory, in: "Classical and Quantum Integrability,'', Banach Center Publ., (2003), 143. Google Scholar [14] J. F. Cariñena, M. F. Rañada and M. Santander, Lagrangian formalism for nonlinear second-order Riccati systems: one-dimensional integrability and two-dimensional superintegrability,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1920287. Google Scholar [15] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Unusual Liénard-type nonlinear oscillator,, Phys. Rev. E, 72 (2005). doi: 10.1103/PhysRevE.72.066203. Google Scholar [16] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 2451. Google Scholar [17] A. G. Choudhury, P. Guha and B. Khanra, Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries,, J. Nonlinear Math. Phys., 15 (2008), 365. doi: 10.2991/jnmp.2008.15.4.2. Google Scholar [18] H. T. Davis., "Introduction to Nonlinear Differential and Integral Equations,'', Dover Publications, (1962). Google Scholar [19] J. M. Dixon and J. A. Tuszyński, Solutions of a generalized Emden equation and their physical significance,, Phys. Rev. A, 41 (1990), 4166. Google Scholar [20] L. Erbe, Comparison theorems for second order Riccati equations with applications,, SIAM J. Math. Anal., 8 (1977), 1032. doi: 10.1137/0508079. Google Scholar [21] V. J. Ervin, W. F. Ames and E. Adams, Nonlinear waves in the pellet fusion process, in:, Wave Phenomena: Modern Theory and Applications, (1984), 199. Google Scholar [22] M. Euler, N. Euler and P. G. L. Leach, The Riccati and Ermakov-Pinney hierarchies,, J. Nonlinear Math. Phys., 14 (2007), 290. doi: 10.2991/jnmp.2007.14.2.10. Google Scholar [23] W. Fair and Y. L. Luke, Rational approximations to the solution of the second order Riccati equation,, Math. Comp., 20 (1966), 602. doi: 10.1090/S0025-5718-1966-0203906-X. Google Scholar [24] R. Flores-Espinoza, Periodic first integrals for Hamiltonian systems of Lie type,, \arXiv{1004.1132}., (). Google Scholar [25] I. A. García, J. Giné and J. Llibre, Liénard and Riccati differential equations related via Lie algebras,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 485. doi: 10.3934/dcdsb.2008.10.485. Google Scholar [26] V. V. Golubev, "Lectures on the Analytical Theory of Differential Equations,'', Gosudarstv. Izdat. Tehn.-Teor. Lit., (1950). Google Scholar [27] A. M. Grundland and D. Levi, On higher-order Riccati equations as Bäcklund transformations,, J. Phys. A, 32 (1999), 3931. doi: 10.1088/0305-4470/32/21/306. Google Scholar [28] A. Guldberg, Sur les équations différentielles ordinaires qui possèdent un système fondamental d'intégrales, (French) [On the differential equations admitting a fundamental system of integrals],, C.R. Math. Acad. Sci. Paris, 116 (1893), 964. Google Scholar [29] N. H. Ibragimov, "Elementary Lie Group Analysis and Ordinary Differential Equations,'', J. Wiley & Sons, (1999). Google Scholar [30] E. L. Ince, "Ordinary Differential Equations,'', Dover Publications, (1944). Google Scholar [31] A. Karasu and P. G. L. Leach, Nonlocal symmetries and integrable ordinary differential equations: $\ddot x + 3x\dot x + x^3 = 0$ and its generalizations,, J. Math. Phys., 50 (2009). Google Scholar [32] S. Lafortune and P. Winternitz, Superposition formulas for pseudounitary matrix Riccati equations,, J. Math. Phys., 37 (1996), 1539. doi: 10.1063/1.531448. Google Scholar [33] J. A. Lázaro-Camí and J. P. Ortega, Superposition rules and stochastic Lie-Scheffers systems,, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 910. Google Scholar [34] S. Lie and G. Scheffers, "Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen,'' (German) [Lectures on continuous groups with geometric (and other) applications],, Teubner, (1893). Google Scholar [35] A. B. Olde Daalhuis, Hyperasymptotics for nonlinear ODEs (II). The first Painlevé equation and a second-order Riccati equation,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3005. Google Scholar [36] M. A. del Olmo, M. A. Rodríguez and P. Winternitz, Simple subgroups of simple Lie groups and nonlinear differential equations with superposition principles,, J. Math. Phys., 27 (1986), 14. doi: 10.1063/1.527381. Google Scholar [37] P. Painlevé, Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, (French) [On second- and higher-order differential equations whose general integral is uniform],, Acta Math., 25 (1902), 1. Google Scholar [38] S. N. Pandey, P. S. Bindu, M. Senthilvelan and M. Lakshmanan, A group theoretical identification of integrable equations in the Liénard-type equation $\ddot x+f(x)+g(x)=0$. II. Equations having maximal Lie point symmetries,, J. Math. Phys., 50 (2009). doi: 10.1063/1.3204075. Google Scholar [39] C. Rogers, W. K. Schief and P. Winternitz, Lie-theoretical generalization and discretization of the Pinney equation,, J. Math. Anal. Appl., 216 (1997), 246. doi: 10.1006/jmaa.1997.5674. Google Scholar [40] C. Tunç and E. Tunç, On the asymptotic behaviour of solutions of certain second-order differential equations,, J. Franklin Inst., 344 (2007), 391. Google Scholar [41] M. E. Vessiot, Sur une classe d'équations différentielles, (French) [On a class of differential equations],, Ann. Sci. École Norm. Sup., 10 (1893), 53. Google Scholar [42] M. E. Vessiot, Sur les systèmes d'équations différentielles du premier ordre qui ont des systèmes fondamentaux d'intégrales, (French) [On the systems of first-order differential equations admitting a fundamental system of integrals],, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894). Google Scholar [43] M. E. Vessiot, Sur quelques équations différentielles ordinaires du second ordre, (French) [On certain second-order differential equations],, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 9 (1895). Google Scholar [44] G. Wallenberg, Sur l'équation différentielle de Riccati du second ordre, (French) [On the second-order Riccati equations],, C.R. Math. Acad. Sci. Paris, 137 (1903), 1033. Google Scholar [45] P. Winternitz, Lie groups and solutions of nonlinear differential equations,, Lecture Notes in Phys., 189 (1983), 263. doi: 10.1007/3-540-12730-5_12. Google Scholar
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