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Details of Grant 

EPSRC Reference: EP/H04261X/1
Title: Integrability in Multidimensions and Boundary Value Problems
Principal Investigator: Fokas, Professor A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Applied Maths and Theoretical Physics
Organisation: University of Cambridge
Scheme: Standard Research
Starts: 01 June 2011 Ends: 31 May 2015 Value (£): 465,254
EPSRC Research Topic Classifications:
Mathematical Analysis Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
04 Mar 2010 Mathematics Prioritisation Panel Deferred
09 Sep 2010 Mathematics Prioritisation Panel Announced
12 May 2010 Mathematics Prioritisation Panel May 2010 Deferred
Summary on Grant Application Form
There exist certain distinctive nonlinear equations called integrable. The impact of the mathematical analysis of such special equations cannot be overestimated. For example, firstly, for integrable equations we have learned detailed aspects of solution behaviour, which includes the long-time asymptotics of solutions and the central role played by solitons. Secondly, it has become apparent that some of the lessons taught by integrable equations have applicability even in non-integrable situations. Indeed, many investigations in the last decade regarding well-posedness of PDEs in appropriate Sobolev spaces have their genesis in results coming from the theory of integrable systems, albeit in non-integrable settings. Perhaps the two most important open problems in the theory of integrable equations have been (a) the solution of initial-boundary as opposed to initial value problems and (b) the derivation and solution of integrable nonlinear PDEs in 3+1 dimensions. In the last 12 years, the PI has made significant progress towards the solution of both of these problems. Namely, regarding (a) he has introduced a unified approach for analysing boundary value problems in two dimensions and regarding (b) he has derived and solved integrable nonlinear PDEs in 4+2. However, many fundamental problems remain open. The proposal aims to investigate several such problems among which the most significant are: (a) the extension of the method for solving boundary value problems from two to three dimensions and (b) the reduction of the new integrable PDEs from 4+2 to 3+1. In addition, the Camassa-Holm analogue of the celebrated sine Gordon equation, several boundary value problems of the elliptic version of the Ernst equation, and the KdV equation on the half-line with time periodic boundary conditions will also be investigated.
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Organisation Website: http://www.cam.ac.uk