Symbolic Software for the Painlevé Test of Nonlinear Ordinary and Partial Differential Equations

References

• Ablowitz , M J and Clarkson , P A . 1991 . Solitons, Nonlinear Evolution Equations and Inverse Scattering , Cambridge, U.K. : Cambridge University Press .
   Google Scholar
• Ablowitz , M J , Ramani , A and Segur , H . 1980 . A Connection Between Nonlinear Evolution Equations and Ordinary Differential Equations of P-type. I . J. Math. Phys. , 21 : 715 – 721 .
   Web of Science ®Google Scholar
• Ablowitz , M J , Ramani , A and Segur , H . 1980 . A Connection Between Nonlinear Evolution Equations and Ordinary Differential Equations of P-type. II . J. Math. Phys. , 21 : 1006 – 1015 .
   Web of Science ®Google Scholar
• Baldwin D, Hereman W, PainleveTest.m: A Mathematica Package for the Painlevé Test of Systems of Nonlinear Ordinary and Partial Differential Equations, 2001; Available at http://www.mines.edu/fs.home/whereman/software/painleve/mathematica (http://www.mines.edu/fs.home/whereman/software/painleve/mathematica.)
   Google Scholar
• Caudrey , P J , Dodd , R K and Gibbon , J D . 1976 . A New Hierarchy of Korteweg-de Vries Equations . Proc. Roy. Soc. Lond. A , 351 : 407 – 422 .
   Web of Science ®Google Scholar
• Clarkson , P A . 1985 . The Painlevé Property and a Partial Differential Equations with an Essential Singularity . Phys. Lett. A , 109 : 205 – 208 .
   Web of Science ®Google Scholar
• Conte , R . 1993 . Singularities of Differential Equations and Integrability, in Introduction to Methods of Complex Analysis and Geometry for Classical Mechanics and Non-Linear Waves , Edited by: Benest , D and Froeschlé , C . 49 – 143 . Gif-sur-Yvette: Editions Frontières .
   Google Scholar
• Conte , R , ed. 1999 . The Painlevé Property: One Century Later, CRM Series in Mathematical Physics , New York : Springer Verlag .
   Google Scholar
• Conte , R , Fordy , A P and Pickering , A . 1993 . A Perturbative Painlevé Approach to Nonlinear Differential Equations . Physica D , 69 : 33 – 58 .
   Google Scholar
• Ercolani , N and Siggia , E D . 1991 . Painlevé Property and Integrability, in What is Integrability? , Edited by: Zakharov , V E . 63 – 72 . Springer Verlag, New York : Springer Series in Nonlinear Dynamics .
   Google Scholar
• Fokas , A S . 1987 . Symmetries and Integrability . Stud. Appl. Math. , 77 : 253 – 299 .
   Web of Science ®Google Scholar
• Fordy , A P and Gibbons , J . 1980 . Some Remarkable Nonlinear Transformations . Phys. Lett. A , 75 : 325 – 325 .
   Web of Science ®Google Scholar
• Goriely , A . 2001 . Integrability and Nonintegrability of Dynamical Systems . Advanced Series in Nonlinear Dynamics , 19 World Scientific Publishing Company, Singapore
   Google Scholar
• Grammaticos , B and Ramani , A . 1997 . Integrability – and How to Detect It, in Integrability of Nonlinear Systems , Edited by: Kosmann-Schwarzbach , Y , Grammaticos , B and Tamizhmani , K . 30 – 94 . Berlin : Springer Verlag .
   Google Scholar
• Hereman , W , Banerjee , P P and Chatterjee , M . 1989 . Derivation and Implicit Solutions of the Harry Dym Equation, and Its Connections with the Korteweg-de Vries Equation . J. Phys. A , 22 : 241 – 255 .
   Google Scholar
• Hereman , W , Göktaş , Ü , Colagrosso , M D and Miller , A J . 1998 . Algorithmic Integrability Tests for Nonlinear Differential and Lattice Equations . Comp. Phys. Comm. , 115 : 428 – 446 .
   Web of Science ®Google Scholar
• Hirota , R and Ramani , A . 1980 . The Miura Transformations of Kaup’s Equation and of Mikhailov’s Equation . Phys. Lett. A , 76 : 95 – 96 .
   Web of Science ®Google Scholar
• Hone , A . 2005 . Painlevé Tests, Singularity Structure, and Integrability . : 1 – 34 . Preprint UKC/IMS/03/33, Institute of Mathematics, Statistics, and Actuarial Science, University of Kent, Canterbury, U.K.; arXiv:nlin.SI/0502017
   Google Scholar
• Ince , E L . 1944 . Ordinary Differential Equations , New York : Dover Publishing Company .
   Google Scholar
• Jimbo , M , Kruskal , M D and Chatterjee , M . 1982 . Painlevé Test for the Self-Dual Yang-Mills Equation . Phys. Lett. A , 92 : 59 – 60 .
   Web of Science ®Google Scholar
• Johnson , R S . 2003 . On Solutions of the Camassa-Holm Equation . Proc. Roy. Soc. Lond. A , 459 : 1687 – 1708 .
   Web of Science ®Google Scholar
• Kruskal , M D and Clarkson , P A . 1992 . The Painlevé-Kowalevski and poly-Painlevé Tests for Integrability . Stud. Appl. Math. , 86 : 87 – 165 .
   Web of Science ®Google Scholar
• Kruskal , M D , Joshi , N and Chatterjee , M . 1997 . Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: A Review and Extensions of Tests for the Painlevé Property, in Proceedings of CIMPA Summer School on Nonlinear Systems , Edited by: Grammaticos , B and Tamizhmani , K . Vol. 495 , 171 – 205 . Heidelberg : Springer Verlag . Lecture Notes in Physics
   Google Scholar
• Lakshmanan , M and Kaliappan , P . 1983 . Lie Transformations, Nonlinear Evolution Equations, and Painlevé Forms . J. Math. Phys. , 24 : 795 – 806 .
   Web of Science ®Google Scholar
• Lamb , G L . 1980 . Elements of Soliton Theory , New York : Wiley and Sons .
   Google Scholar
• Lax , P D . 1968 . Integrals of Nonlinear Equations of Evolution and Solitary Waves . Comm. Pure Appl. Math. , 21 : 467 – 490 .
   Web of Science ®Google Scholar
• McLeod , J B and Olver , P J . 1983 . The Connection Between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painlevé Type . SIAM J. Math. Anal. , 14 : 488 – 506 .
   Web of Science ®Google Scholar
• Newell , A C , Tabor , M and Zeng , Y B . 1987 . A Unified Approach to Painlevé Expansions . Physica D , 29 : 1 – 68 .
   Google Scholar
• Osgood , W F . 1914 . Topics in the Theory of Functions of Several Complex Variables, in The Madison Colloquium 1913 . Colloquium Lectures , 4 : 111 – 230 . AMS, New York
   Google Scholar
• Painlevé , P . 1900 . Mémoire sur les Equations Différentielles dont l’Intégrale Générale est Uniforme . Bull. Soc. Math. France , 28 : 201 – 261 .
   Google Scholar
• Pickering , A . 1996 . The Singular Manifold Method Revisited . J. Math. Phys. , 37 : 1894 – 1927 .
   Web of Science ®Google Scholar
• Ramani , A , Dorizzi , B and Grammaticos , B . 1982 . Painlevé Conjecture Revisited . Phys. Rev. Lett. , 49 : 1539 – 1541 .
   Web of Science ®Google Scholar
• Ramani , A , Grammaticos , B and Bountis , T . 1989 . The Painlevé Property and Singularity Analysis of Integrable and Nonintegrable Systems . Phys. Rep. , 180 : 159 – 245 .
   Web of Science ®Google Scholar
• Rand , D W and Winternitz , P . 1986 . ODEPAINLEVE – A MACSYMA Package for Painlevé Analysis of Ordinary Differential Equations . Comp. Phys. Comm. , 42 : 359 – 383 .
   Web of Science ®Google Scholar
• Renner , F . 1992 . A Constructive REDUCE Package Based Upon the Painlevé Analysis of Nonlinear Evolutions Equations in Hamiltonian and/or Normal Form . Comp. Phys. Comm. , 70 : 409 – 416 .
   Web of Science ®Google Scholar
• Sawada , S and Kotera , T . 1974 . A Method for Finding N-Soliton Solutions of the KdV and KdV-Like Equation . Prog. Theor. Phys. , 51 : 1355 – 1367 .
   Web of Science ®Google Scholar
• Scheen , C . 1997 . Implementation of the Painlevé Test for Ordinary Differential Equations . Theor. Comp. Sci. , 187 : 87 – 104 .
   Web of Science ®Google Scholar
• Steeb , W H and Euler , N . 1988 . Nonlinear Evolution Equations and Painlevé Test , Singapore : World Scientific Publishing Company .
   Google Scholar
• Tabor , M . 1990 . Painlevé Property for Partial Differential Equations, in Soliton Theory: A Survey of Results , Edited by: Fordy , A P . 427 – 446 . Manchester, U.K. : Manchester University Press .
   Google Scholar
• Tan , Y and Yang , J . 2001 . Complexity and Regularity of Vector-Soliton Collisions . Phys. Rev. E , 64 056616
   Web of Science ®Google Scholar
• Ward , R S . 1984 . The Painlevé Property for the Self-dual Gauge-field Equations . Phys. Lett. A , 102 : 279 – 282 .
   Web of Science ®Google Scholar
• Weiss , J . 1983 . The Painlevé Property for Partial Differential Equations. II: Bäcklund Transformations . Lax Pairs, and the Schwarzian Derivative, J. Math. Phys. , 24 : 1405 – 1413 .
   Google Scholar
• Weiss , J . 1984 . Bäcklund Transformation and Linearizations of the Hénon-Heiles System . Phys. Lett. A , 102 : 329 – 331 .
   Web of Science ®Google Scholar
• Weiss , J , Tabor , M and Carnevale , G . 1983 . The Painlevé Property for Partial Differential Equations . J. Math. Phys. , 24 : 522 – 526 .
   Web of Science ®Google Scholar
• Xu , G Q and Li , Z B . 2003 . A Maple Package for the Painlevé Test of Nonlinear Partial Differential Equations . Chin. Phys. Lett. , 20 : 975 – 978 .
   Web of Science ®Google Scholar
• Xu , G Q and Li , Z B . 2004 . Symbolic Computation of the Painlevé Test for Nonlinear Partial Differential Equations Using Maple . Comp. Phys. Comm. , 161 : 65 – 75 .
   Web of Science ®Google Scholar
• Xu , G Q and Li , Z B . 2005 . PDEPtest: A Package for the Painlevé Test of Nonlinear Partial Differential Equations . Appl. Math. Comp. , 169 : 1364 – 1379 .
   Web of Science ®Google Scholar
• Yoshida , H . 1983 . Necessary Condition for the Existence of Algebraic First Integrals I & II . Celes. Mech. , 31 : 363 – 399 .
   Google Scholar
• Ziglin , S L . 1982 . Self-Intersection of the Complex Separatrices and the Nonexistence of the Integrals in the Hamiltonian Systems with One-and-half Degrees of Freedom . J. Appl. Math. Mech. , 45 : 411 – 413 .
   Google Scholar
• Ziglin , S L . 1983 . Branching of Solutions and Nonexistence of First Integrals in Hamiltonian Mechanics I . Func. Anal. Appl. , 16 : 181 – 189 .
   Google Scholar
• Ziglin , S L . 1983 . Branching of Solutions and Nonexistence of First Integrals in Hamiltonian Mechanics II . Func. Anal. Appl. , 17 : 6 – 17 .
   Google Scholar