  Orbital and asymptotic stability for 1-soliton solutions to the Toda lattice
equations as well as small solitary waves to the FPU lattice equations are
established in the energy space. Unlike analogous Hamiltonian PDEs, the lattice
equations do not conserve momentum. Furthermore, the Toda lattice equation is a
bidirectional model that does not fit in with existing theory for Hamiltonian
system by Grillakis, Shatah and Strauss.
  To prove stability of 1-soliton solutions, we split a solution around a
1-soliton into a small solution that moves more slowly than the main solitary
wave, and an exponentially localized part. We apply a decay estimate for
solutions to a linearized Toda equation which has been recently proved by
Mizumachi and Pego to estimate the localized part. We improve the asymptotic
stability results for FPU lattices in a weighted space obtained by Friesecke
and Pego.
