Lie-optics, geometrical phase and nonlinear dynamics of self-focusing and soliton evolution in a plasma

It is useful to state propagation laws for a self-focusing laser beam or a soliton in group-theoretical form to be called Lie-optical form for being a ble to predict self-focusing dynamics conveniently and amongst other things , the geometrical phase. It is shown that the propagation of the gaussian l aser beam is governed by a rotation group in a non-absorbing medium and by the Lorentz group in an absorbing medium if the additional symmetry of para xial propagation is imposed on the laser beam. This latter symmetry, howeve r, needs care in its implementation because the electromagnetic wave of the laser sees a different refractive index profile than the laboratory observ er in this approximation. It is explained how to estimate this non-Taylor p araxial power series approximation. The group theoretical laws so-stated ar e used to predict the geometrical or Berry phase of the laser beam by a tec hnique developed by one of us elsewhere. The group-theoretical Lie-optic (o r ABCD) laws are also useful in predicting the laser behavior in a more com plex optical arrangement like in a laser cavity etc. The nonlinear dynamica l consequences of these laws for long distance (or time) predictions are al so dealt with. Ergodic dynamics of an ensemble of laser beams on the torus during absorptionless self-focusing is discussed in this context. From the point of view of new-physics concepts, we introduce a stroboscopic invarian t torus and a stroboscopic generating function in classical mechanics that is useful for long-distance predictions of absorptionless self-focusing.