# 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), 111-134. 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.
 [1] José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213 [2] Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064 [3] Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339 [4] Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations. Electronic Research Archive, 2021, 29 (6) : 4159-4175. doi: 10.3934/era.2021077 [5] Osama Moaaz, Omar Bazighifan. Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2465-2473. doi: 10.3934/dcdss.2020136 [6] Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001 [7] W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 [8] Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022 [9] Xuan Wu, Huafeng Xiao. Periodic solutions for a class of second-order differential delay equations. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4253-4269. doi: 10.3934/cpaa.2021159 [10] Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053 [11] Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139 [12] Juan Carlos Marrero, David Martín de Diego, Eduardo Martínez. Local convexity for second order differential equations on a Lie algebroid. Journal of Geometric Mechanics, 2021, 13 (3) : 477-499. doi: 10.3934/jgm.2021021 [13] Fasma Diele, Angela Martiradonna, Catalin Trenchea. Stability and errors estimates of a second-order IMSP scheme. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022076 [14] Makoto Okumura, Takeshi Fukao, Daisuke Furihata, Shuji Yoshikawa. A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition. Communications on Pure and Applied Analysis, 2022, 21 (2) : 355-392. doi: 10.3934/cpaa.2021181 [15] Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945 [16] Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299 [17] Maria Do Rosario Grossinho, Rogério Martins. Subharmonic oscillations for some second-order differential equations without Landesman-Lazer conditions. Conference Publications, 2001, 2001 (Special) : 174-181. doi: 10.3934/proc.2001.2001.174 [18] Willy Sarlet, Tom Mestdag. Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations. Journal of Geometric Mechanics, 2022, 14 (1) : 91-104. doi: 10.3934/jgm.2021019 [19] Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609 [20] M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181

2020 Impact Factor: 0.857