scholarly journals A local asymptotic analysis of the discrete first Painlevé equation as the discrete independent variable approaches infinity

1997 ◽  
Vol 4 (2) ◽  
pp. 124-133 ◽  
Author(s):  
Nalini Joshi
Keyword(s):  
Asymptotic Analysis ◽  
Painlevé Equation ◽  
Independent Variable ◽  
Painleve Equation

scholarly journals ON A NOVEL CLASS OF INTEGRABLE ODEs RELATED TO THE PAINLEVÉ EQUATIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250211
Author(s):  
ATHANASSIOS S. FOKAS ◽  
DI YANG
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Independent Variable ◽  
Painlevé Ii ◽  
Painleve Equation

One of the authors recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation h = H(p, q, t), where H is a given Hamiltonian containing t explicitly, yields the function t = T(p, q, h), which defines a new Hamiltonian system with Hamiltonian T and independent variable h. By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlevé I and Painlevé IV equations.

scholarly journals On the asymptotic representation of the solutions to the fourth general Painlevé equation

2003 ◽  
Vol 2003 (13) ◽  
pp. 845-851
Author(s):  
Youmin Lu
Keyword(s):  
Asymptotic Representation ◽  
Painlevé Equation ◽  
Painlevé Transcendents ◽  
Painlevé Transcendent ◽  
Straight Line ◽  
Painleve Transcendents ◽  
Independent Variable ◽  
Painleve Equation

There have been many results on the asymptotics of the Painlevé transcendents in recent years, but the asymptotics of the fourth Painlevé transcendent has not been studied much. In this note, we study the general fourth Painlevé equation and develop an asymptotic representation of a group of its solutions as the independent variable approaches infinity along a straight line.

Sign in / Sign up

Export Citation Format

Share Document