Abstract
We investigate the validity of collective coordinate approximations to the scattering of two solitons in several classes of (1+1) dimensional field theory models. We consider models which are deformations of the sine-Gordon (SG) or the nonlinear Schrodinger (NLS) model which posses soliton solutions (which are topological (SG) or non-topological (NLS)). Our deformations preserve their topology (SG), but change their integrability properties, either completely or partially (models become ‘quasi-integrable’). As the collective coordinate approximation does not allow for the radiation of energy out of a system we look, in some detail, at how the approximation fares in models which are ‘quasi-integrable’ and therefore have asymptotically conserved charges (i.e. charges Q(t) for which Q(t → −∞) = Q(t → ∞)). We find that our collective coordinate approximation, based on geodesic motion etc, works amazingly well in all cases where it is expected to work. This is true for the physical properties of the solitons and even for their quasi-conserved (or not) charges. The only time the approximation is not very reliable (and even then the qualitative features are reasonable, but some details are not reproduced well) involves the processes when the solitons come very close together (within one width of each other) during their scattering.
Highlights
We find that our collective coordinate approximation, based on geodesic motion etc, works amazingly well in all cases where it is expected to work
For the modified nonlinear Schrodinger (NLS) model we are able to return the system to the integrable NLS which we have considered previously by taking our deformation parameter equal to zero; this acts as a check of our numerics, in the calculation of the effective Lagrangian which can be computed analytically only in the case = 0
We have looked at the scattering of solitons in modified NLS models, i.e. models in which the solitons are nontopological and we have demonstrated the usefulness of using our collective coordinate approximation as a tool to investigate their properties
Summary
The idea of using collective coordinates to describe the main features of the scattering of solitons and other extended structures is quite old. It makes sense to approximate the dynamics of slow moving solitons by allowing these parameters to vary in time, i.e. qi = qi(t), and assume that these parameters describe most of the solitons’ dynamics. We start with an approximation ansatz whose form is based on the stationary solution with a suitable choice of parameters, these are taken as collective coordinates and are allowed to depend on time. These coordinates generally describe physical properties of the soliton such as position, height, etc. In our work the equations of motion need to be solved numerically and for this we use a 4th order Runge-Kutta method
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