0 Three Field Dynamics in ( 1 + 1 ) - dimensions

In a model of nonlinear system of three scalar fields the problem on dynamics of a massive particle moving in effective potential provided by two relativistic fields is solving. The potentials for these fields are chosen in the form of anti-Higgs and Higgs potentials. It is shown that the effective potential has the shape of two-hump barrier localized in spacetime. It tends to constant attractive potential at spacetime infinity. The magnitude of this constant constituent is determined by the Higgs condensate. It is shown that nonlinear equation of motion of a particle has the solutions which describe the capture of a particle by the barrier and the scattering on the barrier. 1. Introduction. Study of the dynamics of few fields interacting between themselves is of interest in the connexion with the problems of physics of nonlinear systems which arise in quantum field theory [1], in plasma, in hydrodynamics, in optics and so on [2]. In QCD, for example, if one takes into account spin, colour and flavour degrees of freedom in quark-gluon system the requirement of gauge invariance of the theory leads to clear structure of the Lagrangian [3]. However the equations of motion which follow from this Lagrangian prove to be too complicated for analysis and solution. The modelling of peculiar features of the Lagrangians of few-field systems with nonlinear self-interaction and study on this basis of toy models allowing exact or approximate solutions of the nonlinear equations of motion seem to be important. In the present article nonlinear system of three scalar fields in (1+1)-dimensions is considered. One field is massive while the potentials of two other fields are described by the anti-Higgs and Higgs potentials. In the case when one can separate the slow and rapid motion in spacetime the effective potential of interaction of massive particle with " force field " provided by two relativistic fields is obtained. The mechanism of the formation of this potential is cleared up. It is shown that in general case it has the shape of two-hump barrier localized in spacetime. The potential tends to constant negative value at spacetime infinity. This constant attractive interaction represents the energy determined by the Higgs condensate. It arises in the system as a result of reorganization of vacuum of the field with self-interaction described by the Higgs potential. The equation of motion of massive particle takes into account self-interaction of its field and it …