The scattering of solitons in classes of (1+1) dimensional models

Abstract

We investigate the validity of the collective coordinate approximation to the scattering of two solitons in several classes of (1+1) dimensional field theory models. First we consider the collision of solitons in the integrable NLS model and compare the results of the collective coordinate approximation with results obtained using a full numerical simulation. We find that the approximation is accurate when the solitons are some distance apart and is reasonably good during their interaction.

We then consider a modification of the NLS model with a deformation parameter which changes the integrability properties of the model, either completely or partially (the model becomes quasi-integrable). As the collective coordinate approximation does not allow for the radiation of energy out of a system we pay particular attention to how the approximation fares when the model is quasi-integrable and therefore has asymptotically conserved charges (i.e. charges Q(t) for which Q(t -> -infinity)=Q(t -> infinity)). We find that the approximation accurately reproduces the physical properties of the solitons, and even their anomalous charges, for a large range of initial values. The only time the approximation is not totally reliable is for the scatterings when the solitons come very close together (within one width of each other).

To determine whether these results hold in a model with topological solitons we then consider a modified sine-Gordon model. The deformation preserves the topology of the model but changes the integrability properties in a similar way to the modified NLS model. In this model we find that the approximation is accurate when the model is either integrable or quasi-integrable, but the accuracy was much reduced when the model was completely non-integrable.

To further explore this link between the accuracy of the collective coordinate approximation in a modified sine-Gordon model and the integrability properties of the system, we then consider soliton scattering in a double sine-Gordon model. The double sine-Gordon model allows us to vary between two integrable sine-Gordon models, and when the model is not integrable it still possesses the additional symmetries necessary for quasi-integrability. We find that for all values of our deformation parameters the approximation is accurate and that, as expected, the anomalous charges are asymptotically conserved.