scholarly journals Truncation-type methods and Bäcklund transformations for ordinary differential equations: the third and fifth Painlevé equations

2001 ◽  
Vol 43 (A) ◽  
pp. 23-32 ◽  
Author(s):  
P. R. Gordoa ◽  
N. Joshi ◽  
A. Pickering
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Painlevé Equations ◽  
Bäcklund Transformations ◽  
Painleve Equations ◽  
The Third ◽  
Backlund Transformations

scholarly journals Solutions of the second and fourth Painlevé equations, I

1997 ◽  
Vol 148 ◽  
pp. 151-198 ◽  
Author(s):  
Hiroshi Umemura ◽  
Humihiko Watanabe
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Painlevé Equations ◽  
Necessary Condition ◽  
Hamiltonian Vector Field ◽  
Rigorous Proof ◽  
Painleve Equations ◽  
Algebraic Differential Equations ◽  
The Third ◽  
Principal Ideals

AbstractA rigorous proof of the irreducibility of the second and fourth Painlevé equations is given by applying Umemura’s theory on algebraic differential equations ([26], [27], [28]) to the two equations. The proof consists of two parts: to determine a necessary condition for the parameters of the existence of principal ideals invariant under the Hamiltonian vector field; to determine the principal invariant ideals for a parameter where the principal invariant ideals exist. Our method is released from complicated calculation, and applicable to the proof of the irreducibility of the third, fifth and sixth equation (e.g. [32]).

scholarly journals Bäcklund transformations for several cases of a type of generalized KdV equation

2004 ◽  
Vol 2004 (63) ◽  
pp. 3369-3377
Author(s):  
Paul Bracken
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Generalized Kdv Equation ◽  
Backlund Transformations ◽  
De Vries ◽  
Derivative Term

An alternate generalized Korteweg-de Vries system is studied here. A procedure for generating solutions is given. A theorem is presented, which is subsequently applied to this equation to obtain a type of Bäcklund transformation for several specific cases of the power of the derivative term appearing in the equation. In the process, several interesting, new, ordinary, differential equations are generated and studied.

scholarly journals Painleve Test and Discrete Boltzmann Equations

1989 ◽  
Vol 42 (1) ◽  
pp. 1 ◽  
Author(s):  
N Euler ◽  
W-H Steeb
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Lie Symmetry ◽  
Bäcklund Transformations ◽  
Painlevé Analysis ◽  
Painlevé Test ◽  
Boltzmann Equations ◽  
Backlund Transformations ◽  
Painleve Analysis

The Painleve test for various discrete Boltzmann equations is performed. The connection with integrability is discussed. Furthermore the Lie symmetry vector fields are derived and group-theoretical reduction of the discrete Boltzmann equations to ordinary differentiable equations is performed. Lie Backlund transformations are gained by performing the Painleve analysis for the ordinary differential equations

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