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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
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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