Symmetry, singularities and integrability in complex dynamics. I: The reduction problem. (English) Zbl 0989.34074
The authors investigate five systems of first-order ordinary differential equations (for 2, 3, 4 functions) with quadratic right-hand side, which has been discussed in a former paper of I. Z. Golubchik and V. V. Sokolov [J. Nonlinear Math. Phys. 7, No. 2, 184-197 (2000; Zbl 1119.37318)].
Several ways connected with the Painlevé test and symmetry analysis to get reductions as well as solutions and integrability (in various versions) are outlined in detail for every system.
Several ways connected with the Painlevé test and symmetry analysis to get reductions as well as solutions and integrability (in various versions) are outlined in detail for every system.
Reviewer: G.Czichowski (Greifswald)
MSC:
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |
Citations:
Zbl 1119.37318Software:
DIMSYMReferences:
| [1] | Abraham-Shrauner, B.; Govinder, KS; Leach, PGL, Integration of Second Order Equations Not Possessing Lie Point Symmetries, PhysLett A, 203, 168-174, 1995 · Zbl 1020.34500 |
| [2] | Burgan J.-R., Sur des groupes de transformation en physique mathématique: Application aux fluides de éspace des phases et à la méchanique quantique, PhD thesis, Université d’Orléans, Orléans, 1978. |
| [3] | Burgan J.-R., Munier A., Fijkalow E. and Feix M.R., Groupes de redimensionnement et transformations canoniques générlisées en Physique non-lineaire, Problèmes Inverses, Sabatier P C ed, Cahiers Mathématiques, Montpellier, 1982, 67-76. |
| [4] | Cairó, L.; Feix, MR; Hua, DD; Bouquet, S.; Dewisme, A., Hamiltonian Method and Invariant Search for 2D Quadratic Systems, J PhysA: Math Gen, 26, 4371-4386, 1993 · Zbl 0813.34007 |
| [5] | Cairó, L.; Feix, MR; Géronimi, C., On the Integrability of the Three-Dimensional Lotka-Volterra System, 66-71, 1998 |
| [6] | Cairó, L.; Llibre, J.; Feix, MR, Darboux Method and Search for Invariants of the Lotka-Volterra and Complex Quadratic Systems, J Math Phys, 40, 2074-2091, 1999 · Zbl 0986.34032 |
| [7] | Chishlom, JSR; Common, AK, A Class of Second Order Differential Equations and Related First Order Systems, J PhysA:Math Gen, 20, 5459-5472, 1987 · Zbl 0631.34046 |
| [8] | Conte, R., Singularities of Differential Equations and Integrability, 49-143, 1994, Gif-sur-Yvette: Editions Frontières, Gif-sur-Yvette · Zbl 0948.34065 |
| [9] | Contopoulos, G.; Grammaticos, B.; Ramani, A., Painlevé Analysis for the Mixmaster Model, J PhysA: Math Gen, 26, 5795-5799, 1993 · Zbl 0811.53086 |
| [10] | Contopoulos, G.; Grammaticos, B.; Ramani, A., The Mixmaster Universe Revisited, J Phys A: Math Gen, 27, 5357-5362, 1994 · Zbl 0844.58120 |
| [11] | Spiros, C.; Leach, PGL, Painlevé Analysis of the Mixmaster Universe, J PhysA: Math Gen, 27, 1625-1631, 1994 · Zbl 0843.58144 |
| [12] | Erwin, VJ, Ames W.F. and Adams E., 1984, Amsterdam: North Holland, Amsterdam |
| [13] | Feix, MR; Géronimi, C.; Cairó, L.; Leach, PGL; Lemmer, RL; Bouquet, SÉ, On the Singularity Analysis of Ordinary Differential Equations Invariant under Time Translation and Rescaling, J PhysA: Math Gen, 30, 7437-7461, 1997 · Zbl 0927.34006 |
| [14] | Claude, Géronimi; Feix Marc, R.; Leach Peter, GL, Periodic Solutions, Limit Cycles and the Generalised Chazy Equation, 327-335, 1999, Heildelberg: Springer-Verlag, Heildelberg |
| [15] | Golubev, VV, Lectures on Analytical Theory of Differential Equations, 1950, Moscow-Leningrad: Gostekhizdat, Moscow-Leningrad · Zbl 0038.24201 |
| [16] | Golubchik, IZ; Sokolov, VV, On Some Generalisations of the Factorisation Method, Theoret and Math Phys, 110, 267-276, 1997 · Zbl 0922.58034 |
| [17] | Golubchik, IZ; Sokolov, VV, Operator Yang-Baxter Equations, Integrable ODEs and Nonassociative Algebras, J Nonlin Math Phys, 7, 184-197, 2000 · Zbl 1119.37318 |
| [18] | Head, AK, LIE, a PC Program for Lie Analysis of Differential Equations, Comp Phys Commun, 77, 241-248, 1993 · Zbl 0854.65055 |
| [19] | Hua, DD; Cairó, L.; Feix, MR; Govinder, KS; Leach, PGL, Connection Between the Existence of First Integrals and the Painlevé Property in Lotka-Volterra and Quadratic Systems, Proc Roy Soc, 452, 859-880, 1996 · Zbl 0873.34005 |
| [20] | Kamke, E., Differentialgleichungen: Lösungsmethoden und Lösungen, Band I, 10th Aufl, B.G, 1983, Stuttgart: Teubner, Stuttgart |
| [21] | Kostant, B., The Solution to a Generalised Toda Lattice and Representation Theory, Adv Math, 34, 195-338, 1979 · Zbl 0433.22008 |
| [22] | Krause, J., On the Complete Symmetry Group of the Classical Kepler System, J Math Phys, 35, 5734-5748, 1994 · Zbl 0816.70006 |
| [23] | Leach, PGL, First Integrals for the Modified Emden Equation \(\ddot q + \alpha (t)\dot q + {q^n} = 0\), J Math Phys, 26, 2510-2514, 1985 · Zbl 0587.34004 |
| [24] | Leach, PGL, Hierarchies of Similarity Symmetries and Singularity Analysis, 304-312, 1999, Heidelberg: Springer-Verlag, Heidelberg · Zbl 1229.34018 |
| [25] | Leach P.G.L., Cotsakis S. and Flessas G.P., Symmetry, Singularities and Integrability in Complex Dynamics II: Rescaling and Time-Translation in Two-Dimensional Systems, 2000, (to appear in J Math Anal Appln). · Zbl 0992.34027 |
| [26] | Leach P.G.L., Nucci M.C. and Cotsakis S., Symmetry, Singularities and Integrability in Complex Dynamics III: Complete Symmetry Groups of Nonintegrable Ordinary Differential Equations, 2000, (submitted to J Nonlin Math Phys). |
| [27] | Lemmer, RL; Leach, PGL, The Painlevé Test, Hidden Symmetries and the Equation y″ + yy′ + ky3 = 0, J Phys A, 26, 5017-5024, 1993 · Zbl 0806.34008 |
| [28] | Lie Sophus, Differentialgleichungen, Teubner, Leipsig, 1891, (reprinted: Chelsea, New York, 1967). |
| [29] | Maharaj, SD; Leach, PGL, Exact Solutions for the Tikekar Superdense Star, J Math Phys, 37, 430-437, 1996 · Zbl 0873.34004 |
| [30] | Mahomed, FM; Leach, PGL, The Linear Symmetries of a Nonlinear Differential Equation, Quæst Math, 8, 241-274, 1985 · Zbl 0618.34009 |
| [31] | Mahomed, FM; Leach, PGL, Contact Symmetries of Second Order Differential Equations, J Math Phys, 32, 2051-2055, 1991 · Zbl 0738.34008 |
| [32] | Moreira, IC, Exact Polynomial Invariants for Newtonian Systems, Had J, 7, 475-492, 1984 · Zbl 0545.70027 |
| [33] | Sherring, J.; Head, AK; Prince, GE, Dimsym and LIE: Symmetry Determining Packages, Math Comp Model, 25, 153-164, 1997 · Zbl 0918.34007 |
| [34] | Tabor, M., Chaos and Integrability in Nonlinear Dynamics, 1989, New York: John Wiley, New York · Zbl 0682.58003 |
| [35] | Tikekar, R., Exact Model for a Relativistic Star, J Math Phys, 31, 2454-2458, 1990 · Zbl 0726.70011 |
| [36] | Tikekar Ramesh, A Class of Exact Solutions of Einstein Equations in Higher Dimensional Space-Times, J Indian Math Soc, 1995, V.61, 37-45. · Zbl 0853.53080 |
| [37] | Vaidya, PC; Tikekar, R., Exact Relativistic Models for a Superdense Star, J Astrophys Astr, 3, 325-334, 1982 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
