We exactly solve a class of Frenkel-Kontorova models with a periodic potential composed of piecewise convex parabolas having the same curvature. All rotationally ordered stable configurations can be depicted with appropriate phase parameters. The elements of a phase parameter are grouped into subcommensurate clusters. Phase transitions, manifested in the gap structure changes previously seen in numerical simulations, correspond to the splitting and merging of subcommensurate clusters with the appearance of incommensurate nonrecurrent rotationally ordered stable configurations. Through the notion of elementary phase shifts, all the possibilities for the existence of configurations degenerate with the ground state are scrutinized and the domains of stability in the phase diagram are characterized. At the boundaries of a domain of stability, nonrecurrent minimum energy configurations are degenerate with the ground state configurations and phase transitions occur.
Physical Review B (Condensed Matter and Materials Physics) 66(18), 184108(22pages)