2 - Differential equations featuring many periodic solutions

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.