Further investigations in solitons

Description

The idea of this project is to pick up from where the solitons course left off. In particular we will begin with many of the features you discovered for the KdV equation (soliton solutions, Lax Pair, hierarchy etc.) and consider more general equations sharing these festures. A particular generalisation is to the KP equation. The KP equation (proposed by Kadomtsev and Petviashvili in 1970) is a non-linear PDE describing shallow water wave fronts eg how wave fronts arriving at angles to each other (eg on a beach) interact.

It is a two dimensional generalisation of the KdV equation. Like the KdV equation it is integrable and has solitons solutions. Indeed it acts as a universal equation for many other integrable systems. More recently it has been found that the solutions are related to the "positive Grassmannian": matrices with positive minors (submatrices) which are related to a number of beautiful mathematical areas (cluster algebras, plabic diagrams etc.) and also (where my own interest lie) in turn have been recently related to scattering amplitudes, relevant for processes currently studies in the Large Hadron Collider in CERN.

In this project you will study the KP equation (and possibly other generalisations eg the 2d Toda lattice), following and generalising many of the features you examined for the KdV equation. There will be the opportunity to go in a number of different directions from here. One idea would be to study the positive Grassmannian and the relation to the KP equation which will leave room to go in various further directions, either in a more pure direction - cluster algebras, plabic diagrams etc. - or a more physical one, ie applications to scattering amplitudes.

Prerequisites

Solitons III. This project very much continues from where that module finished.

Resources and references

Checkout the wikipedia, and scholarpedia pages for the KP equation and the links there. The book "Glimpses of Soliton Theory" by Alex Kasman will be very useful. There are many introductions and lectures to various apsects of this but this set of lectures may be good to start with.