Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 Introduction
- 2 Differential equations featuring many periodic solutions
- 3 Geometry and integrability
- 4 The anti-self-dual Yang–Mills equations and their reductions
- 5 Curvature and integrability for Bianchi-type IX metrics
- 6 Twistor theory for integrable systems
- 7 Nonlinear equations and the ∂̅-problem
2 - Differential equations featuring many periodic solutions
Published online by Cambridge University Press: 06 January 2010
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 Introduction
- 2 Differential equations featuring many periodic solutions
- 3 Geometry and integrability
- 4 The anti-self-dual Yang–Mills equations and their reductions
- 5 Curvature and integrability for Bianchi-type IX metrics
- 6 Twistor theory for integrable systems
- 7 Nonlinear equations and the ∂̅-problem
Summary
Abstract
A simple trick is reviewed, which yields differential equations (both ODEs and PDEs) of evolution type featuring lots of periodic solutions. Several examples (PDEs) are exhibited.
Introduction
Recently a simple trick has been introduced that allows us to manufacture evolution equations (both ODEs and PDEs) which possess lots of periodic solutions – in particular, completely periodic solutions corresponding, in the context of the initial-value problem, to an open set of initial data of nonvanishing measure in the space of initial data. The purpose and scope of this presentation is to review this trick – most completely introduced and described in – and to display, and tersely discuss, certain new (classes of) evolution PDEs yielded by it; the alert reader, after having grasped the main idea, can easily manufacture many more examples, possibly also featuring several dependent and independent variables – here for simplicity we restrict attention to just one (complex) dependent variable and to just two (real) independent variables (the standard 1 + 1 case: one ‘time’ and one ‘space’ variables only).
The trick is described tersely in Section 2.2. Some examples of evolution equations – different from those reported in – are displayed in Section 2.3, which should be immediately seen by the browser who wishes to decide whether to invest time in reading the rest of this paper.
- Type
- Chapter
- Information
- Geometry and Integrability , pp. 9 - 20Publisher: Cambridge University PressPrint publication year: 2003
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