Classical waves on nonlinear lattices

Abstract

We discuss the solution to classical vibrations on several nonlinear lattices in one dimension. One lattice has nearest-neighbor potential energy with both quadratic and quartic terms in the relative displacements $q$. Another lattice has the potential energy terms going as $\mathrm{cosh}(q)$. Exact analytical solutions are derived for periodic waves that have a period of two, three, and four lattice constants. Several of these cases employ Jacobian elliptic functions, while one solution uses ordinary cosines. The quadratic term in the potential energy can have either sign, and a double well occurs when it is negative and the quartic is positive. Solutions are also found for this double-well potential.