Details of Grant 

EPSRC Reference:	GR/M09841/01
Title:	SOLITONS ON RIEMANN SURFACES AND LATTICES
Principal Investigator:	Speight, Professor JM
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Researcher Co-Investigators:
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Department:	Pure Mathematics
Organisation:	University of Leeds
Scheme:	Postdoc Res Fellowship PreFEC
Starts:	01 October 1999	Ends:	30 September 2001	Value (£):	58,728
EPSRC Research Topic Classifications:	Mathematical Physics Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Summary on Grant Application Form
The main goal of the proposed research is to provide rigorous justification for the geodesic approximation to lump dynamics in the CP1 model, by adapting the vortex analysis of Stuart. The model on a compact Riemann surface is well suited to this for several reasons: first, the geodesic approximation is free of singularities, second, the spectral theory of Hessian of the potential energy, central to the analysis, is simpler on a compact surface and third, the n-lump moduli space Mn is well understood, being the space of degree n meromorphic functions on i. In the geodesic approximation, Mn is equipped with a Riemannian structure g, about whose geometry very little is known. It is known that (Mn, g) is geodesically incomplete and two low degree, low genus examples have been studied in detail, but the general case remains little understood. A second goal is to prove the following conjectures, all motivated by known examples: (a) that (Mn, g) is Kahler, (b) that (Mn, g) has finite diameter and finite volume, and (c) that the end of (Mn, g) has codimension 2. Conjectures (b) and (c) will require good asymptotic estimates of g towards the end of Mn, where the lumps are narrow and spiky.The third goal is somewhat different: to study breathers in highly discrete systems. In particular, one would like to understand the direction of continuation of breathers from the uncoupled limit, and the mechanism of breakdown of continuation. This work will be partly analytic and partly numerical.
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Organisation Website:	http://www.leeds.ac.uk