Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach

@article{Clarkson1998BcklundTF,
  title={B{\"a}cklund transformations for the second Painlev{\'e} hierarchy: a modified truncation approach},
  author={Peter A. Clarkson and Nalini T. Joshi and Andrew Pickering},
  journal={Inverse Problems},
  year={1998},
  volume={15},
  pages={175-187}
}
The second Painlev? hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well known second Painlev? equation, . In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlev? analysis to ordinary differential equations. We extend these techniques in order to derive auto-B?cklund transformations for the second Painlev… 

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References

SHOWING 1-10 OF 30 REFERENCES

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

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

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations

Rational solutions for Schwarzian integrable hierarchies

We give an approach to finding rational solutions of completely intagrable hierarchies, which makes use of the relationship between modifications and the Schwarzian equations obtained via the

Lax pairs, bäcklund transformations and special solutions for ordinary differential equations

The authors investigate a modification of the Weiss-Tabor-Carnevale procedure that enables one to obtain Lax pairs and Backlund transformations for systems of ordinary differential equations. This

On a unified approach to transformations and elementary solutions of Painlevé equations

An algorithmic method is developed for investigating the transformation properties of second‐order equations of Painleve type. This method, which utilizes the singularity structure of these

Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations

Given a partial differential equation, its Painleve analysis will first be performed with a built‐in invariance under the homographic group acting on the singular manifold function. Then, assuming an

Monodromy- and spectrum-preserving deformations I

A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential