Unstable Manifolds and Soliton Dynamics

Abstract

It is argued that the interactions between solitons are well approximated by a finite-dimensional dynamical system, provided that the static forces between the solitons are relatively weak. The application to skyrmion dynamics and the nucleon-nucleon interaction is described.PACS numbers: 1 l. lO.Lm, 02.40.+m, 13.75.Cs This Letter is concerned with the truncation of a Lagrangean field theory to a finite-dimensional Lagrangean dynamical system. Most classical field theories, and all those considered here, are based on an infinite-dimensional configuration space G, the space of field configurations at a particular time, on which is defined a metric and a potential energy function V. From the metric one obtains the kinetic energy T as an infinite-dimensional version of jgijV l v J . The Lagrangean of the theory is T -V. Lorentz invariance implies that the functional forms of T and V are related, but this will not concern us. A truncation of the theory is defined as the Lagrangean dynamics on a submanifold M of G, with the induced metric and potential energy function. In general, the solutions of the equations of motion on M will not correspond to solutions of the original field equations, because the original theory has no forces constraining the motion to M. In some circumstances, however, there are effective constraining forces, and the full field dynamics does approximately reduce to motion on a suitable finite-dimensional manifold M. Such a truncation is particularly useful in theories with solitons-meaning vortices, monopoles, skyrmions, etc., rather than the solitons in exactly integrable theories. An approximate quantization is possible with Hamiltonian -V 2 +F, where V 2 is the covariant Laplacian on M and V the induced potential.In the dynamics of single solitons one usually makes a finite-dimensional truncation of the infinite-dimensional field dynamics. A single soliton configuration is never unique and is labeled by collective coordinates which include position coordinates, and often internal coordinates as well. These collective coordinates parametrize a manifold M which is an orbit of the symmetry group of the theory. The metric on M is invariant under the group action and depends on the mass and moments of inertia of the soliton, and also on moments associated with internal symmetries. The potential energy of M is a constant, equal to the mass of the soliton, so the Lagrangean on M has essentially just a kinetic term coming from the metric. The dynamics on M is therefore geodesic motion at constant speed. This accurately describes the classical motion of a free soliton provided all speeds are small. The geodesic dynamics gives non-Abelian monopoles spatial momentum and electric charge. It gives skyrmions a correlated angular momentum and isospin as well as spatial momentum. The geodesic dynamics is only an approximation, as it ignores Lorentz contraction and the centrifugal deformation of the soliton due to internal motion.A second situation where one can truncate the field theory is in the low-ene...

Recent mathematical developments in the Skyrme model

“…The problem could be tractable if there were some way of truncating to a nite number of degrees of freedom. Exactly such a truncation was suggested by Manton [37].…”

“…We expect the relevant l o w-energy space of con gurations to also have 12 dimensions. Manton [37] proposed that this sub-manifold could be obtained as the union of all gradient o w curves linking together all the low-energy critical points. We will return to this subject in much detail in section 4.…”

“…linear, theory of elasticity. De ne G the displacement gradient G m = ( A I) m = @u @X m : (37) Then one can de ne two tensors of strain:…”

“…We start by presenting in section 4.1 Manton's method [37] for truncating the degrees of freedom of a system, thereby possibly rendering it tractable. This formalism was published by Manton after his work on BPS monopole (topological solitons of the massless SU(2) Higgs model) scattering [38].…”

A Skyrme model approach to the spin-orbit force

“…The usual procedure for quantising one Skyrmion is to use a rigid body approach to the classical minimal energy configuration, promoting its collective coordinates to quantum operators. Quantising the interaction between separated Skyrmions is more difficult and little progress has been possible using the full set of collective coordinates [11]. Thus, we will only consider a toy model in two dimensions where we treat the Skyrmions as rigid discs.…”