Painlevé Tests, Singularity Structure and Integrability

@article{Hone2009PainlevTS,
  title={Painlev{\'e} Tests, Singularity Structure and Integrability},
  author={Andrew N W Hone},
  journal={arXiv: Exactly Solvable and Integrable Systems},
  year={2009},
  pages={245-277}
}
  • A. Hone
  • Published 9 February 2005
  • Mathematics
  • arXiv: Exactly Solvable and Integrable Systems
After a brief introduction to the Painleve property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painleve tests. The tests are applied to several different examples, and the connection between singularity structure and integrability of ordinary and partial differential equations is discussed. 

Singularity Analysis and Integrability of a Burgers-Type System of Foursov

We apply the Painleve test for integrability of partial differential equations to a system of two coupled Burgers-type equations found by Foursov, which was recently shown by Sergyeyev to possess

Painlevé Test to a Reduced System of Six Coupled Nonlinear ODEs

Abstract: In this paper we investigate the complete integrability of the system of six coupled nonlinear ODEs, which arises in the ODE reduction of rotating stratified Boussinesq equations. We use

On Painlevé analysis for some non–linear evolution equations

In this paper, we present explicit Painlevé test for the potential Boussinesq equation, The murrary equation, The (2 + 1) Calogero equation, The Rosenau – Hyman equation (RH), Cole – Hopf (CH)

Lax Representation and Darboux Solutions of the Classical Painlevé Second Equation

In this article, we present Darboux solutions of the classical Painlevé second equation. We reexpress the classical Painlevé second Lax pair in new setting introducing gauge transformations to yield

The Painlevé Tests, Bäcklund Transformation and Bilinear Form for the KdV Equation with a Self-Consistent Source

The Painleve property and Backlund transformation for the KdV equation with a self-consistent source are presented. By testing the equation, it is shown that the equation has the Painleve property.

Lax pair representation and Darboux transformation of NC Painlevé-II equation

The extension of Painleve equations to noncommutative spaces has been considering extensively in the theory of integrable systems and it is also interesting to explore some remarkable aspects of

On Integrability of Christou’s Sixth Order Solitary Wave Equations

  • M. Allami
  • Mathematics
    Iraqi Journal of Science
  • 2019
We examine the integrability in terms of Painlevè analysis for several models of higher order nonlinear solitary wave equations which were recently derived by Christou. Our results point out that

Integrability study of a four-dimensional eighth-order nonlinear wave equation

We study the integrability of the four-dimensional eighth-order nonlinear wave equation of Kac and Wakimoto, associated with the exceptional affine Lie algebra ${\mathfrak e}_6^{(1)}$. Using the
...

References

SHOWING 1-10 OF 138 REFERENCES

The Painlevé property for partial differential equations

In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing

Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property

The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is

THE PAINLEVE PROPERTY FOR PARTIAL DIFFERENTIAL EQUATIONS. II. BACKLUND TRANSFORMATION, LAX PAIRS, AND THE SCHWARZIAN DERIVATIVE

In this paper we investigate the Painleve property for partial differential equations. By application to several well‐known partial differential equations (Burgers, KdV, MKdV, Bousinesq, higher‐order

Bilinear discrete Painleve equations

Based on the results of singularity confinement we derive bilinear expressions for the discrete Painleve equations. In these cases where a bilinear expression is not sufficient we obtain trilinear or

Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations

The integrability of systems of ordinary differential equations with polynomial vector fields is investigated by using the singularity analysis methods. Three types of results are obtained. First, a

On Painlevé's equations I, II and IV

We give a new proof of the fact that the solutions of Painlevé's differential equations I, II and IV are meromorphic functions in the complex plane. The method of proof is based on differential

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations
...