Lie point symmetries and ODEs passing the Painlevé test

Abstract

The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlevé property are explored for ODEs of order n = 2, . . . , 5. Among the 6 ODEs identifying the Painlevé transcendents only PIII , PV and PV I have nontrivial symmetry algebras and that only for very special values of the parameters. In those cases the transcendents can be expressed in terms of simpler functions, i.e. elementary functions, solutions of linear equations, elliptic functions or Painlevé transcendents occurring at lower order. For higher order or higher degree ODEs that pass the Painlevé test only very partial classifications have been published. We consider many examples that exist in the literature and show how their symmetry groups help to identify those that may define genuinely new transcendents.

Symmetries and hamiltonians of Ince’s XXXVIII and XLIX equations

“…These equations presented in Ince's book [4] are solvable, meaning that their solutions are expressible in terms of known functions. Comparing with literature on Painlevé equations, the Hamiltonian structure and symmetries of solvable equations on Ince's list attracted much less attention with notable exceptions of few recent publications [5,6,7]. The present study fills this gap by studying symmetries of equations I 38 and I 49 : from Ince's list [4,5,6,7].…”

“…Comparing with literature on Painlevé equations, the Hamiltonian structure and symmetries of solvable equations on Ince's list attracted much less attention with notable exceptions of few recent publications [5,6,7]. The present study fills this gap by studying symmetries of equations I 38 and I 49 : from Ince's list [4,5,6,7]. For the full understanding of their symmetries it is instructive to first study how their structures emerge in the context of P III−V model [3,1].…”

Similarity transformations and linearization for a family of dispersionless integrable PDEs

“…Another important application of the Lie symmetry approach is the classification scheme of differential equations according to the admitted group of symmetries, and to the construction of equivalent transformation which transform a given differential equation into another differential equation of the same order, when the admitted Lie symmetries form the same Lie algebra [33][34][35]. Recently, in [36] the authors investigated which of the six ordinary differential equations of the Painlevé transcendents admit nontrivial Lie point symmetries. It was found that equations P III , P V and P V I have nontrivial symmetries for special values of the free parameters.…”

Similarity Transformations and Linearization for a Family of Dispersionless Integrable PDEs

“…Another important application of the Lie symmetry approach is the classification scheme of differential equations according to the admitted group of symmetries, and to the construction of equivalent transformation which transform a given differential equation into another differential equation of the same order, when the admitted Lie symmetries form the same Lie algebra [33][34][35]. Recently, in [36] the authors investigated which of the six ordinary differential equations of the Painlevé transcendents admit nontrivial Lie point symmetries. It was found that equations P I I I , P V and P V I have nontrivial symmetries for special values of the free parameters.…”