Prolongations, Bäcklund Transformations, and Lie Theory as Algorithms for Solving and Understanding Nonlinear Differential Equations

1978 ◽  
pp. 33-60
Author(s):  
Robert Hermann
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Bäcklund Transformations ◽  
Lie Theory ◽  
Backlund Transformations

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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