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 $2times2$-matrix Lame 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 Schrodinger 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.
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