Wave stability on one-dimensional non-linear lattices

We report the results of our stability analysis of exact travling wave solutions for two non-linear mono-atomic lattices in one dimension. One lattice has nearest-neighbor potential energy containg quadratic and quartic terms (Fermi-Pasta-Ulam model). The other lattice has potential energy which goes as $cosh(q)$, a generalization of the Toda lattice. These exact traveling wave solutions have wave lengths that are commensurate with the lattice constant. It is found that on the quadratic-quartic lattice, the traveling wave solutions are unstable. For the $cosh(q)$ lattice, on the other hand, the solutions are stable.