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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 |
References:
[1] |
C. Arnold, Formal continued fractions solutions of the generalized second order Riccati equations, applications, Numer. Algorithms, 15 (1997), 111-134.
doi: 10.1023/A:1019262520178. |
[2] |
J. Beckers, L. Gagnon, V. Hussin and P. Winternitz, Superposition formulas for nonlinear superequations, J. Math. Phys., 31 (1990), 2528-2534.
doi: 10.1063/1.528997. |
[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), 1480-1490.
doi: 10.1063/1.529306. |
[4] |
J. F. Cariñena, J. Grabowski and J. de Lucas, Quasi-Lie schemes: theory and applications, J. Phys. A, 42 (2009), 335206.
doi: 10.1088/1751-8113/42/33/335206. |
[5] |
J. F. Cariñena, J. Grabowski and J. de Lucas, Lie families: theory and applications, J. Phys A, 43 (2010), 305201.
doi: 10.1088/1751-8113/43/30/305201. |
[6] |
J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258. |
[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), 012009.
doi: 10.1088/1742-6596/175/1/012009. |
[8] |
J. F. Cariñena, P. G. L. Leach and J. de Lucas, Quasi-Lie systems and Emden-Fowler equations, J. Math. Phys., 50 (2009), 103515. |
[9] |
J. F. Cariñena and J. de Lucas, A nonlinear superposition rule for solutions of the Milne-Pinney equation, Phys. Lett. A, 372 (2008), 5385-5389. |
[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), 304029.
doi: 10.1088/1751-8113/41/30/304029. |
[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: http://www.springerlink.com/content/51mu8h705025m158/. |
[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), 031. |
[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., 59, Polish Acad. Sci., Warsaw, (2003), 143-162. |
[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), 062703.
doi: 10.1063/1.1920287. |
[15] |
V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Unusual Liénard-type nonlinear oscillator, Phys. Rev. E, 72 (2005), 066203.
doi: 10.1103/PhysRevE.72.066203. |
[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), 2451-2476. |
[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), 365-382.
doi: 10.2991/jnmp.2008.15.4.2. |
[18] |
H. T. Davis., "Introduction to Nonlinear Differential and Integral Equations,'' Dover Publications, New York, 1962. |
[19] |
J. M. Dixon and J. A. Tuszyński, Solutions of a generalized Emden equation and their physical significance, Phys. Rev. A, 41 (1990), 4166-4173. |
[20] |
L. Erbe, Comparison theorems for second order Riccati equations with applications, SIAM J. Math. Anal., 8 (1977), 1032-1037.
doi: 10.1137/0508079. |
[21] |
V. J. Ervin, W. F. Ames and E. Adams, Nonlinear waves in the pellet fusion process, in: "Wave Phenomena: Modern Theory and Applications," North Holland Mathematics Studies, 97, Amsterdam, (1984), 199-210. |
[22] |
M. Euler, N. Euler and P. G. L. Leach, The Riccati and Ermakov-Pinney hierarchies, J. Nonlinear Math. Phys., 14 (2007), 290-310.
doi: 10.2991/jnmp.2007.14.2.10. |
[23] |
W. Fair and Y. L. Luke, Rational approximations to the solution of the second order Riccati equation, Math. Comp., 20 (1966), 602-606.
doi: 10.1090/S0025-5718-1966-0203906-X. |
[24] |
R. Flores-Espinoza, Periodic first integrals for Hamiltonian systems of Lie type,, \arXiv{1004.1132}., ().
|
[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), 485-494.
doi: 10.3934/dcdsb.2008.10.485. |
[26] |
V. V. Golubev, "Lectures on the Analytical Theory of Differential Equations,'' Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950. |
[27] |
A. M. Grundland and D. Levi, On higher-order Riccati equations as Bäcklund transformations, J. Phys. A, 32 (1999), 3931-3937.
doi: 10.1088/0305-4470/32/21/306. |
[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), 964-965. |
[29] |
N. H. Ibragimov, "Elementary Lie Group Analysis and Ordinary Differential Equations,'' J. Wiley & Sons, Chichester, 1999. |
[30] |
E. L. Ince, "Ordinary Differential Equations,'' Dover Publications, New York, 1944. |
[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), 073509. |
[32] |
S. Lafortune and P. Winternitz, Superposition formulas for pseudounitary matrix Riccati equations, J. Math. Phys., 37 (1996), 1539-1550.
doi: 10.1063/1.531448. |
[33] |
J. A. Lázaro-Camí and J. P. Ortega, Superposition rules and stochastic Lie-Scheffers systems, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 910-931. |
[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, Leipzig, 1893. |
[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), 3005-3021. |
[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), 14-23.
doi: 10.1063/1.527381. |
[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), 1-85. |
[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), 102701.
doi: 10.1063/1.3204075. |
[39] |
C. Rogers, W. K. Schief and P. Winternitz, Lie-theoretical generalization and discretization of the Pinney equation, J. Math. Anal. Appl., 216 (1997), 246-264.
doi: 10.1006/jmaa.1997.5674. |
[40] |
C. Tunç and E. Tunç, On the asymptotic behaviour of solutions of certain second-order differential equations, J. Franklin Inst., 344 (2007), 391-398. |
[41] |
M. E. Vessiot, Sur une classe d'équations différentielles, (French) [On a class of differential equations], Ann. Sci. École Norm. Sup., 10 (1893), 53-64. |
[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), H1-H33. |
[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), F1-F26. |
[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), 1033-1035. |
[45] |
P. Winternitz, Lie groups and solutions of nonlinear differential equations, Lecture Notes in Phys., 189 (1983), 263-305.
doi: 10.1007/3-540-12730-5_12. |
show all references
References:
[1] |
C. Arnold, Formal continued fractions solutions of the generalized second order Riccati equations, applications, Numer. Algorithms, 15 (1997), 111-134.
doi: 10.1023/A:1019262520178. |
[2] |
J. Beckers, L. Gagnon, V. Hussin and P. Winternitz, Superposition formulas for nonlinear superequations, J. Math. Phys., 31 (1990), 2528-2534.
doi: 10.1063/1.528997. |
[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), 1480-1490.
doi: 10.1063/1.529306. |
[4] |
J. F. Cariñena, J. Grabowski and J. de Lucas, Quasi-Lie schemes: theory and applications, J. Phys. A, 42 (2009), 335206.
doi: 10.1088/1751-8113/42/33/335206. |
[5] |
J. F. Cariñena, J. Grabowski and J. de Lucas, Lie families: theory and applications, J. Phys A, 43 (2010), 305201.
doi: 10.1088/1751-8113/43/30/305201. |
[6] |
J. F. Cariñena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem, and partial differential equations, Rep. Math. Phys., 60 (2007), 237-258. |
[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), 012009.
doi: 10.1088/1742-6596/175/1/012009. |
[8] |
J. F. Cariñena, P. G. L. Leach and J. de Lucas, Quasi-Lie systems and Emden-Fowler equations, J. Math. Phys., 50 (2009), 103515. |
[9] |
J. F. Cariñena and J. de Lucas, A nonlinear superposition rule for solutions of the Milne-Pinney equation, Phys. Lett. A, 372 (2008), 5385-5389. |
[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), 304029.
doi: 10.1088/1751-8113/41/30/304029. |
[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: http://www.springerlink.com/content/51mu8h705025m158/. |
[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), 031. |
[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., 59, Polish Acad. Sci., Warsaw, (2003), 143-162. |
[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), 062703.
doi: 10.1063/1.1920287. |
[15] |
V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, Unusual Liénard-type nonlinear oscillator, Phys. Rev. E, 72 (2005), 066203.
doi: 10.1103/PhysRevE.72.066203. |
[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), 2451-2476. |
[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), 365-382.
doi: 10.2991/jnmp.2008.15.4.2. |
[18] |
H. T. Davis., "Introduction to Nonlinear Differential and Integral Equations,'' Dover Publications, New York, 1962. |
[19] |
J. M. Dixon and J. A. Tuszyński, Solutions of a generalized Emden equation and their physical significance, Phys. Rev. A, 41 (1990), 4166-4173. |
[20] |
L. Erbe, Comparison theorems for second order Riccati equations with applications, SIAM J. Math. Anal., 8 (1977), 1032-1037.
doi: 10.1137/0508079. |
[21] |
V. J. Ervin, W. F. Ames and E. Adams, Nonlinear waves in the pellet fusion process, in: "Wave Phenomena: Modern Theory and Applications," North Holland Mathematics Studies, 97, Amsterdam, (1984), 199-210. |
[22] |
M. Euler, N. Euler and P. G. L. Leach, The Riccati and Ermakov-Pinney hierarchies, J. Nonlinear Math. Phys., 14 (2007), 290-310.
doi: 10.2991/jnmp.2007.14.2.10. |
[23] |
W. Fair and Y. L. Luke, Rational approximations to the solution of the second order Riccati equation, Math. Comp., 20 (1966), 602-606.
doi: 10.1090/S0025-5718-1966-0203906-X. |
[24] |
R. Flores-Espinoza, Periodic first integrals for Hamiltonian systems of Lie type,, \arXiv{1004.1132}., ().
|
[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), 485-494.
doi: 10.3934/dcdsb.2008.10.485. |
[26] |
V. V. Golubev, "Lectures on the Analytical Theory of Differential Equations,'' Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950. |
[27] |
A. M. Grundland and D. Levi, On higher-order Riccati equations as Bäcklund transformations, J. Phys. A, 32 (1999), 3931-3937.
doi: 10.1088/0305-4470/32/21/306. |
[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), 964-965. |
[29] |
N. H. Ibragimov, "Elementary Lie Group Analysis and Ordinary Differential Equations,'' J. Wiley & Sons, Chichester, 1999. |
[30] |
E. L. Ince, "Ordinary Differential Equations,'' Dover Publications, New York, 1944. |
[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), 073509. |
[32] |
S. Lafortune and P. Winternitz, Superposition formulas for pseudounitary matrix Riccati equations, J. Math. Phys., 37 (1996), 1539-1550.
doi: 10.1063/1.531448. |
[33] |
J. A. Lázaro-Camí and J. P. Ortega, Superposition rules and stochastic Lie-Scheffers systems, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 910-931. |
[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, Leipzig, 1893. |
[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), 3005-3021. |
[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), 14-23.
doi: 10.1063/1.527381. |
[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), 1-85. |
[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), 102701.
doi: 10.1063/1.3204075. |
[39] |
C. Rogers, W. K. Schief and P. Winternitz, Lie-theoretical generalization and discretization of the Pinney equation, J. Math. Anal. Appl., 216 (1997), 246-264.
doi: 10.1006/jmaa.1997.5674. |
[40] |
C. Tunç and E. Tunç, On the asymptotic behaviour of solutions of certain second-order differential equations, J. Franklin Inst., 344 (2007), 391-398. |
[41] |
M. E. Vessiot, Sur une classe d'équations différentielles, (French) [On a class of differential equations], Ann. Sci. École Norm. Sup., 10 (1893), 53-64. |
[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), H1-H33. |
[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), F1-F26. |
[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), 1033-1035. |
[45] |
P. Winternitz, Lie groups and solutions of nonlinear differential equations, Lecture Notes in Phys., 189 (1983), 263-305.
doi: 10.1007/3-540-12730-5_12. |
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