The Cubic - quintic Schrödinger equation and
the generalized coherent states
M. Aguero G.
Universidad Autonoma del Estado de Mexico,
Facultad de Ciencias, Toluca,
Apartado Postal 2-139, Edo de Mexico, Mexico
The quassiclassical description of quantum systems were studied widely in
the literature. To study collective excitations (magnons) over
ground states, one should procced from the quantum level of description to
a quasiclassical one, and such a transition should be done very carefully.
Thus the problem to formulate a consistent reduction procedure appears.
In fact this procedure consists in choosing trial functions (i.e. some
basis) which can be used for averaging the hamiltonian. It is natural to
choose for that coherent states (CS) since these states are the most
classical and minimize the uncertainty relation acording Perelomov.
Recent discoveries in Coherent states are reviewed in light of their
possible implications for nonlinear models and related subjects in
theoretical physics [1]. The recent discoveries of interest here are of
various surprising ``exotic'' Behaviour. The path to such structures
intertwines many branches of mathematics and theoretical physics
(Integrable systems and other gauge theories). An overview of these topics
is provided, including a brief review of some of the less familiar
mathematics. This is followed by discussion of certain results concerning
the application of the Generalized Coherent States in the study of
dynamics of the Phi -six model in the classical nonlinear field theories.
The Genralized Coherent Sates are defined as a points of the factor spaces
. The investigations provide us with
remarkable properties of the soliton solutions in both models and the
phase transitions is analized from a more elegant perspective. The
possible implication of chaotic behaviour is analysed in the lattice
version of this model. Other suggestions and conjectures for future
research are made.