Instanton Moduli and Topological Soliton Dynamics

It has been proposed by Atiyah and Manton that the dynamics of Skyrmions may be approximated by motion on a finite dimensional manifold obtained from the moduli space of SU(2) Yang-Mills instantons. Motivated by this work we describe how similar results exist for other soliton and instanton systems. We describe in detail two examples for the approximation of the infinite dimensional dynamics of sine-Gordon solitons by finite dimensional dynamics on a manifold obtained from instanton moduli. In the first example we use the moduli space of CP1 instantons and in the second example we use the moduli space of SU(2) Yang-Mills instantons. The metric and potential functions on these manifolds are constructed and the resulting dynamics is compared with the explicit exact soliton solutions of the sine-Gordon theory.Comment: uuencoded tex file, 27 pages including 4 figures, requires phyzzx macro. DAMTP 94-5

R-adaptive multisymplectic and variational integrators

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this paper we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. Numerical results for the Sine-Gordon equation are also presented.Comment: 65 pages, 13 figure