A unified approach to Painlevé expansions

Research output: Contribution to journalArticle

124 Citations (Scopus)

Abstract

The Painlevé test for partial differential equations developed by Weiss, Tabor and Carnevale (WTC) is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations. A simple modification of the WTC procedure used for partial differential equations enables us to determine the Lax pairs, Hirota equations and auto-Bäcklund transformations for ordinary differential equations, including a new Lax pair for an integrable case of the Henon-Heiles system. A detailed study of the KdV hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained. The role of the singular branches is examined in detail and important new insights obtained. In particular we find that each singular branch is simply a re-expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce. A parallel analysis is carried out for the AKNS hierarchy.

Original languageEnglish (US)
Pages (from-to)1-68
Number of pages68
JournalPhysica D: Nonlinear Phenomena
Volume29
Issue number1-2
DOIs
StatePublished - 1987

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partial differential equations
Partial differential equations
Branch
Ordinary differential equations
hierarchies
expansion
Partial differential equation
Lax Pair
Poles
poles
Pole
Ordinary differential equation
KdV Hierarchy
differential equations

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

A unified approach to Painlevé expansions. / Newell, Alan C; Tabor, Michael; Zeng, Y. B.

In: Physica D: Nonlinear Phenomena, Vol. 29, No. 1-2, 1987, p. 1-68.

Research output: Contribution to journalArticle

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