  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.
