Nonlinear systems: algebraic structures and integrability

The research proposed concerns methods for solving nonlinear partial differential equations. Such equations are used to model many sophisticated phenomena in the natural/physical world. The goals of such models are to make predictions, find optimal solutions or maybe even to control outcomes. Being able to find exact or sufficiently accurate solutions efficiently is crucial to these enterprises. However, finding such solutions to nonlinear equations is notoriously difficult. A now famous classical method for finding exact solutions to some classes of such nonlinear equations, known as the Inverse Scattering Transform, has been around since the sixties. The method essentially breaks the solution process into solving a combination of two linear equations: a linear integral equation known as the Gel'fand-Levitan-Marchenko equation and a linearised version of the nonlinear partial differential equation concerned. This approach generates the famous soliton solutions, of the Korteweg-de Vries equation modelling shallow water waves, and of the nonlinear Schrodinger equation modelling pulse propagation in optical fibres. Such equations are said to be "integrable".

The proposer's recent research has shone some new light on this classical solution procedure, in two ways. First, that a simplified version of the procedure readily generates solutions to large classes of nonlocal nonlinear partial differential equations, including in particular, specific classes of coagulation equations. Such equations model cluster formation such as in blood clotting or polymerisation or nanoparticle surface deposition. Second, by abstracting the solution procedure, the proposer has shown how the integrability of the Korteweg-de Vries and nonlinear Schrodinger equations is equivalent to establishing the existence of polynomial expansions in an associated combinatorial algebra. The research proposed herein seeks to extend these results along these two directions, to: (i) Demonstrate how a variation on the simplified procedure can be used to determine solutions to general classes of coagulation equations and use this in some of the applications indicated; and (ii) Extend the abstract procedure to include all the main known classical integrable equations, as well as use it to establish new integrable equations by starting to classify the possible systems that fit within the abstract framework developed. Projects (i) and (ii) represent completely new science. The proposer will also begin to look to establish connections between these procedures and solution representations for such nonlinear equations based on random processes. The intention is to submit a consequential larger grant based on the results established herein.