Aspects of Integrability of Differential Systems and Fields

TL;DR

This monograph introduces the concept of integrability in differential systems and fields, covering methods for solving ordinary and partial differential equations, with discussions on advanced topics like Lax pairs.

Contribution

It provides an educational overview of integrability concepts, including path-independence, first integrals, and geometric methods, with insights into advanced techniques such as Lax pairs.

Findings

Comprehensive introduction to integrability in differential systems.

Exploration of methods for integrating ODEs and PDEs.

Discussion of advanced topics like Bäcklund transformations.

Abstract

This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases of integrability examined are: path-independence of line integrals of vector fields on the plane and in space; integration of systems of ordinary differential equations by using first integrals; and integrable systems of partial differential equations. Special topics include the integration of analytic functions on the complex plane and some elements from the geometric theory of differential systems. Certain more advanced subjects, such as Lax pairs and B\"acklund transformations, are also discussed.