Singularity Analysis and a Function Unifying the Painlevé and the Psi Series

References

• Ablowitz , M J , Ramani , A and Segur , H . 1978 . Nonlinear Evolution Equations and Ordinary Differential Equations of Painlevé Type . Lett. Nuovo Cimento , 23 : 333 – 337 .
   Google Scholar
• Ablowitz , M R , 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
• Bureau , F J . 1964 . Differential Equations with Fixed Critical Points . Annali di Matematica pura e applicata , LXVI : 1 – 116 .
   Google Scholar
• Chazy , J . 1911 . Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes . Thèse Paris (1910), Acta Math. , 34 : 317 – 385 .
   Google Scholar
• Conte R Singularities of Differential Equations and Integrability, in Introduction to Methods of Complex Analysis and Geometry for Classical Mechanics and Nonlinear Waves, Editors: Benest D and Frœschlé C, Éditions Frontières, Gif-sur-Yvette, 1993
   Google Scholar
• Cosgrove , C M and Scoufis , G . 1993 . Painlevé Classification of a Class of Differential Equations of the Second Order and Second Degree . Stud. Appl. Math. , 88 : 25 – 87 .
   Web of Science ®Google Scholar
• Davis , H T . 1962 . Introduction to Nonlinear Differential and Integral Equations , New York : Dover .
   Google Scholar
• Ervin V I Ames W J Adams E Nonlinear Waves and Pellet Fusion, in Wave Phenomena: Modern Theory and Applications, Editors: Rodgers C and Moody J B North Holland, Amsterdam, 1984
   Google Scholar
• Feix , M R , Géronimi , C , Cairó , L , Leach , P G L , Lemmer , R L and Bouquet , S É . 1997 . Right and Left Painlevé Series for Ordinary Differential Equations Invariant Under Time Translation and Rescaling . J. Phys. A: Math. Gen. , 30 : 7437 – 7461 .
   Google Scholar
• Fordy , A and Pickering , A . 1991 . Analysing Negative Resonances in the Painlevé Test . Phys. Lett. , A160 : 347 – 354 .
   Google Scholar
• Géronimi C Feix M R Leach P G L Third Order Differential Equation Possessing Three Symmetries (the Two Homogeneous Ones Plus the Time Translation), Preprint: MAPMO/URA CNRS 1803 – Université d’Orléans, Département mathématiques BP 6759, 45067 La Source, Orléans Cedex (2)France, 1997
   Google Scholar
• Géronimi C Feix M R Leach P G L Periodic solutions and Associated Limit Cycles for the Generalised Chazy Equation, in Dynamical Systems, Plasmas and Gravitation, Editors: Leach P G L, Bouquet S É, Rouet J-L and Fijalkow E, Springer-Verlag, Heildelberg, 1999, 327–335
   Google Scholar
• Golubev VV Lectures on Analytical Theory of Differential Equations, Gosteekhizdat, Moscow – Leningrad, 1950
   Google Scholar
• Hua , D D , Cairó , L , Feix , M R , Govinder , K S and Leach , P G L . 1996 . Connection between the Existence of First Integrals and the Painlevé Property in Lotka–Volterra and Quadratic Systems . Proc. Roy. Soc. , 452 : 859 – 880 .
   Google Scholar
• Ince , E L . 1927 . Ordinary Differential Equations , London : Longmans, Green and Co .
   Google Scholar
• Leach , P G L . 1985 . First Integrals for the Modified Emden Equation . J. Math. Phys. , 26 : 2510 – 2514 .
   Web of Science ®Google Scholar
• Leach , P G L , Feix , M R and Bouquet , S . 1988 . Analysis and Solution of a Nonlinear Second Order Differential Equation through Rescaling and through a Dynamical Point of View . J. Math. Phys. , 29 : 2563 – 2569 .
   Web of Science ®Google Scholar
• Mahomed , F M and Leach , P G L . 1985 . The Linear Symmetries of a Nonlinear Differential Equation . Quaestiones Mathematicae , 8 : 241 – 274 .
   Google Scholar
• Lemmer , R L and Leach , P G L . 1993 . The Painlevé Test, Hidden Symmetries and the Equation y″+yy′+ky 3=0 . J. Phys. A: Math. Gen. , 26 : 5017 – 5024 .
   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