2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/153832We solve exactly a class of Frenkel-Kontorova models with piecewise parabolic potential, which has $d$ sub-wells in a period. With careful analysis, we show that the phase diagram of the minimum enthalpy configurations exhibits the structure of a complete $d$-dimensional devil's staircase. The winding number of a minimum enthalpy configuration is locked to rational values, while the fraction of atoms in each sub-well is locked to values which are sub-commensurable with the winding number.14 pages, uses revtex. Four figures includedExactly Solvable and Integrable Systemstext