Description

Most interesting physical systems which have forces or interactions in them are described by nonlinear differential equations. Unfortunately in general such equations cannot be solved analytically, and one has to resort to approximation techniques such as perturbation theory to extract useful information. However there exist some rare exceptions to this rule, so called integrable models, which despite being nonlinear, are `exactly' solvable. This is on account of their rich underlying symmetry; they have as many conserved quantities as degrees of freedom, all of whose Poisson brackets vanish with one another. These models are also connected with the phenomenon of integrable solitons. Solitons are `localised energy' solutions to PDE's, which do not decay even when scattered off one another. As such they are fascinating as a sort of classical model of particles.

The aim of this project will be to provide an overview of integrable models by studying some basic examples. These could range from models with a finite number of degrees of freedom, to more sophisticated field theory models such as Toda theories and the nonlinear Schrodinger equation. Many beautiful methods, such as Backlund transformations (a sort of fancy version of the Cauchy-Riemann conditions), Lax pairs, and inverse scattering can be used to study these models.

Prerequisites

Resources

• Drazin and Johnson; Solitons an Introduction
• Lamb: Elements of Soliton Theory
• More books on Solitons in the Library
• Perelemov: Integrable systems of classical mechanics and lie algebras

• Dunajski: Integrable Systems