  This paper concerns the existence of generalized solitary waves (solitary
waves with small ripples at infinity) for a diatomic Fermi-Pasta-Ulam-Tsingou
(FPUT) lattice. It is proved that the FPUT lattice problem has a generalized
solitary-wave solution with the amplitude of those ripples algebraically small
using dynamical system approach. The problem is first formulated as a dynamical
system problem and then the center manifold reduction theorem with Laurent
series expansion is applied to show that this system can be reduced to a system
of ordinary differential equations with dimension five. Its dominant system has
a homoclinic solution. By applying a perturbation method and adjusting some
appropriate constants, it is shown that this homoclinic solution persists for
the original dynamical system, which connects to a periodic solution of
algebraically small amplitude at infinity (called generalized homoclinic
solution), which yields the existence of a generalized solitary wave for the
FPUT lattice. The result presented here with the algebraic smallness of those
ripples will be needed to show the existence of generalized multi-hump waves
for the FPUT lattice later.
