Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice

Abstract

The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and $n$ particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated in this short note. For each $k$ dividing $n$ we find $k$ degree of freedom invariant manifolds. They represent short wavelength solutions composed of $k$ Fourier-modes and can be interpreted as embedded lattices with periodic boundary conditions and only $k$ particles. Inside these invariant manifolds other invariant structures and exact solutions are found which represent for instance periodic and quasi-periodic solutions and standing and traveling waves. Some of these results have been found previously by other authors via a study of mode coupling coefficients and recently also by investigating `bushes of normal modes'. The method of this paper is similar to the latter method and much more systematic than the former. We arrive at previously unknown results without any difficult computations. It is shown moreover that similar invariant manifolds exist also in the Klein-Gordon lattice and in the thermodynamic and continuum limits.