Farey Tree and the Frenkel-Kontorova Model

We solved the Frenkel-Kontorova model with the potential $V(u)= -\frac{1}{2} |\lambda|(u-{\rm Int}[u]-\frac{1}{2})^2$ exactly. For given $|\lambda|$, there exists a positive integer $q_c$ such that for almost all values of the tensile force $\sigma$, the winding number $\omega$ of the ground state configuration is a rational number in the $q_c$-th level Farey tree. For fixed $\omega=p/q$, there is a critical $\lambda_c$ when a first order phase transition occurs. This phase transition can be understood as the dissociation of a large molecule into two smaller ones in a manner dictated by the Farey tree. A kind of ``commensurate-incommensurate'' transition occurs at critical values of $\sigma$ when two sizes of molecules co-exist. ``Soliton'' in the usual sense does not exist but induces a transformation of one size of molecules into the other.