  Soliton crystals are periodic patterns of multi-spot optical fields formed
from either time or space entanglements of equally separated identical
high-intensity pulses. These specific nonlinear optical structures have gained
interest in recent years with the advent and progress in nonlinear optical
fibers and fiber lasers, photonic crystals, wave-guided wave systems and most
recently optical ring microresonator devices. In this work an extensive
analysis of characteristic features of soliton crystals is carried out, with
emphasis on their one-to-one correspondance with Elliptic solitons. In this
purpose we examine their formation, their stability and their dynamics in
ring-shaped nonlinear optical media within the framework of the Lugiato-Lefever
equation. The stability analysis deals with internal modes of the system via a
$2\times2$-matrix Lam\'e type eigenvalue problem, the spectrum of which is
shown to possess a rich set of boundstates consisting of stable zero-fequency
modes and unstable decaying as well as growing modes. Turning towards the
dynamics of Elliptic solitons in ring-shaped fiber resonators with Kerr
nonlinearity, first of all we propose a collective-coordinate approach, based
on a Lagrangian formalism suitable for Elliptic-soliton solutions to the
nonlinear Schr\'odinger equation with an arbitrary perturbation. Next we derive
time evolutions of Elliptic-soliton parameters in the specific context of
ring-shaped optical fiber resonators, where the optical field evolution is
tought to be governed by the Lugiato-Lefever equation. By solving numerically
the collective-coordinate equations an analysis of the amplitude, the position,
the phase of internal oscillations, the phase velocity and the energy is
carried out and reveals a complex dynamics of the Elliptic soliton in
ring-shaped optical microresonators.
