The Connection between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painleve Type.

Abstract

A completely integrable partial differential equation is one which has a Lax representation, or, more precisely, can be solved via a linear integral equation of Gel'fand-Levitan type, the classic example being the Korteweg-de Vries equation. An ordinary differential equation is of Painleve type if the only singularities of its solutions in the complex plane are poles. It is shown that, under certain restrictions, if G is an analytic, regular symmetry group of a completely integrable partial differential equation, then the reduced ordinary differential equation for the G-invariant solutions is necessarily of Painleve type. This gives a useful necessary condition for complete integrability, which is applied to investigate the integrability of certain generalizations of the Korteweg-de Vries equation, Klein-Gordon equations, some model nonlinear wave equations of Whitham and Benjamin, and the BBM equation. (Author)

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Document Details

Document Type
Technical Report
Publication Date
Nov 01, 1980
Accession Number
ADA093616

Entities

People

  • J. B. Mcleod
  • P. J. Olver

Organizations

  • University of Wisconsin–Madison

Tags

Communities of Interest

  • Air Platforms

DTIC Thesaurus Topics

  • Banach Space
  • Differential Equations
  • Electrical Solitons
  • Equations
  • Formulas (Mathematics)
  • Integral Equations
  • Integrals
  • Inverse Scattering
  • Mathematics
  • New York
  • Nonlinear Differential Equations
  • Partial Differential Equations
  • Scattering
  • Solitons
  • United States
  • Wave Equations
  • Waves

Fields of Study

  • Mathematics

Readers

  • Calculus or Mathematical Analysis
  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Linear Algebra