  The dynamical stability of solitary lattice waves in non-integrable FPUT
chains is a longstanding open problem and has been solved so far only in a
certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in
which the waves propagate with near sonic speed, have large wave length, and
carry low energy. In this paper we derive a similar result in a complementary
asymptotic regime related to fast and strongly localized waves with high
energy. In particular, we show that the spectrum of the linearized FPUT
operator contains asymptotically no unstable eigenvalues except for the neutral
ones that stem from the shift symmetry and the spatial discreteness. This
ensures that high-energy waves are linearly stable in some orbital sense, and
the corresponding nonlinear stability is granted by the general, non-asymptotic
part of the seminal Friesecke-Pego result and the extension by Mizumachi.
  Our analytical work splits into two principal parts. First we refine
two-scale techniques that relate high-energy wave to a nonlinear asymptotic
shape ODE and provide accurate approximation formulas. In this way we establish
the existence, local uniqueness, smooth parameter dependence, and exponential
localization of fast lattice waves for a wide class of interaction potentials
with algebraic singularity. Afterwards we study the crucial eigenvalue problem
in exponentially weighted spaces, so that there is no unstable essential
spectrum. Our key argument is that all proper eigenfunctions can asymptotically
be linked to the unique bounded and normalized solution of the linearized shape
ODE, and this finally enables us to disprove the existence of unstable
eigenfunctions in the symplectic complement of the neutral ones.
