266
Views
17
CrossRef citations to date
0
Altmetric
Original Articles

The Riccati and Ermakov-Pinney hierarchies

, &
Pages 290-310 | Published online: 21 Jan 2013

References

  • Abraham-Shrauner , B , Leach , PGL , Govinder , KS and Ratcliff , G . 1995 . Hidden and contact symmetries of ordinary differential equations . J. Phys. A , 23 : 6707 – 6716 .
  • Cavalcante , J A and Tenenblat , K . 1988 . Conservation Laws for Nonlinear Evolution Equations . J. Math. Phys. , 29 : 1044 – 1049 .
  • Conte , R . 1994 . Singularities of differential equations and integrability, in Introduction to Methods of Complex Analysis and Geometry for Classical Mechanics and Nonlinear Waves , Edited by: Benest , D and Frceschlé . 49 – 143 . Gif-sur-Yvette : Éditions Frontiéres .
  • Ermakov , V . 1880 . Second-order differential equations. Conditions of complete integrability . Universita Izvestia Kiev Series III , 9 : 1 – 25 . trans. Harin A O
  • Euler , M , Euler , N and Petersson , N . 2003 . Linearizable hierarchies of evolution equations in (1 + 1) dimensions . Stud. Appl. Math. , 111 : 315 – 337 .
  • Feix , M R , Géronimi , C , Cairó , Leach , P G L , Lemmer , R L and Bouquet , S E . 1997 . On the singularity analysis of ordinary differential equations invariant under time translation and rescaling . J. Phys. A , 30 : 7437 – 7461 .
  • Fokas , A S . 1980 . A symmetry approach to exactly solvable evolution equations . J. Math. Phys. , 21 : 1318 – 1325 .
  • Fokas , A S . 1987 . Symmetries and integrability . Stud. Appl. Math. , 77 : 253 – 299 .
  • Fuchssteiner , B . 1979 . Application of hereditary symmetries to nonlinear evolution equations . Nonlinear Anal.-Theory Methods Appl. , 3 : 849 – 862 .
  • Géronimi , C , Leach , P G L and Feix , M R . 2002 . Singularity analysis and a function unifying the Painlevé and the Psi series . J. Nonlinear Math. Phys. , 9 ( 2 ) : 36 – 48 .
  • Gorringe , V M and Leach , P G L . 1993 . The first integrals and their Lie algebra of the most general autonomous Hamltonian of the form H = T + V possessing a Laplace-Runge-Lenz vector . J. Aust. Math. Soc. B , 34 : 511 – 522 .
  • Hall , L S . 1982 . A theory of exact and configurational invariants . Physica D , 8 : 90 – 113 .
  • Ince , E L . 1927 . Ordinary Differential Equations , Green & Co , , London : Longmans .
  • Leach , P G L , Cotsakis , S and Flessas , G P . 2000 . Symmetry, singularities and integrability in complex dynamics I: The reduction problem . J. Nonlinear Math. Phys. , 7 : 445 – 479 .
  • Mahomed , F M and Leach , P G L . 1990 . Symmetry Lie algebras of nth order ordinary differential equations . J. Math. Anal. Appl. , 151 : 80 – 107 .
  • Morozov , V V . 1958 . Classification of six-dimensional nilpotent Lie algebras . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 5 : 161 – 171 .
  • Mubarakzyanov , G M . 1963 . On solvable Lie algebras . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 32 : 114 – 123 .
  • Mubarakzyanov , G M . 1963 . Classification of real structures of five-dimensional Lie algebras . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 34 : 99 – 106 .
  • Mubarakzyanov , G M . 1963 . Classification of solvable six-dimensional Lie algebras with one nilpotent base element . Izvestia Vysshikh Uchebn Zavendeniĭ Matematika , 35 : 104 – 116 .
  • Olver , P J . 1977 . Evolution equations possessing infinitely many symmetries . J. Math. Phys. , 18 : 1212 – 1215 .
  • Painlevé , P . 1902 . Sur les équations difiérentielles du second ordre et d’ordre supérieur dont l’intégrale générale est uniforme . Acta Math. , 25 : 1 – 86 .
  • Petersson , N , Euler , N and Euler , M . 2004 . Recursion operators for a class of integrable third-order evolution equations . Stud. Appl. Math. , 112 : 201 – 225 .
  • Pinney , E . 1950 . The nonlinear differential equation y˝(x) + p(x)y + cy−3 = 0 . Proc. Amer. Math. Soc. , 1 : 681
  • Rajasekar , S . 2004 . Painlevé analysis and integrability of two coupled nonlinear oscillators . Pramana - J. Phys. , 62 : 1 – 12 .
  • Ramani , A , Grammaticos , B and Bountis , T . 1989 . The Painlevé Property and singularity analysis of integrable and nonintegrable systems . Phys. Rep. , 180 : 159 – 245 .
  • Riccati , J F . 1724 . Acta Eruditorum quæ Lipsiae publicantur Supplementa , VIII : 74
  • Sarlet , W , Leach , PGL and Cantrijn , F . 1985 . Exact versus configurational invariants and a weak form of complete integrability . Physcia D , 17 : 331 – 338 .
  • Sen , T . 1987 . A class of integrable potentials . J. Math. Phys. , 28 : 2841 – 2850 .
  • Tabor , M . 1989 . Chaos and Integrability in Nonlinear Dynamics , New York : John Wiley .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.