# 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…

## 117 Citations

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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

- Mathematics
- 1983

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

- Mathematics
- 1983

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

- Mathematics
- 1999

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

- Mathematics
- 1998

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

- Physics
- 1988

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

- Mathematics
- 1982

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

- Mathematics
- 1991

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

- Mathematics
- 1980

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…