Details of Grant 

EPSRC Reference:	GR/A00133/01
Title:	AF: MATHEMATICAL ANALYSIS OF SOLITONS IN CLASSICAL FIELD THEORY
Principal Investigator:	Stuart, Dr DMA
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Department:	Applied Maths and Theoretical Physics
Organisation:	University of Cambridge
Scheme:	Advanced Fellowship (Pre-FEC)
Starts:	01 April 2000	Ends:	31 December 2003	Value (£):	131,291
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Summary on Grant Application Form
The central focus of this proposal is to develop rigorous analytical techniques for studying the dynamical behaviour of solitons in non-integrable classical field theories. The anticipated methodology is that of the theory of PDE's combined with techniques from geometry, asymptotic analysis and dynamical systems. Examples of particular interest are superconductivity (Ginzburg-Landau theory), the Yang-Mills-Higgs equations and general relativity. It is initially expedient to restrict attention to various regimes in which a simplified description is possible. This involves the study of solutions of PDE's in various limits as some small parameter vanishes. In the adiabatic limit the small parameter is the value of the (conserved) energy above the minimum value (in a given topological class); projection onto the space of equilibria then gives, in certain cases, a useful, valid approximation to the original dynamical system. I intend to develop a general formulation of this procedure for infinite dimensional Hamiltonian systems which includes the known interesting examples. In the particle limit the small parameter is the ratio of the size of the soliton to other length scales in the problem. A particularly important problem of the latter type is the motion of test particles in general relativity: they are expected to follow geodesics in the background space-time but there is as yet no satisfactory analytic treatment of this problem available. It is proposed here to study this when the matter Lagrangian supports soliton solutions; the test-particle can then be modelled by a soliton. The first stage is to consider soliton equations on a space-time with fixed background metric and construct solutions in which the soliton concentrates along a geodesic. The next stage is to couple the soliton equation to Einstein's equation so that the metric becomes dynamical; this is more difficult as it will require estimates of the effect of the soliton on the metric (in the test-particle limit). Another case of interest is the large-time limit; an important open problem is to understand the large-time behaviour of (and develop a scattering theory for) nonlinear classical field theories in which solitons are present. An optimistic working conjecture is that as t > +oo the solution decomposes into a collection of uniformly moving solitons (and possibly bound states of solitons) plus a radiation field which behaves in an essentially linear fashion. Also it is proposed to study various other related questions such as: the existence of time-periodic solutions representing bound states of solitons (breathers), in particular the existence of solutions with superconducting vortices rotating about one another; regularity of the Yang-Mills equations in critical dimension and the role of instantons in possible blow-up; existence of monopoles in the Einstein-Yang-Mills-Higgs system and related heat flow problems.
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