  Using a nonlocal version of the center manifold theorem and a normal form
reduction, we prove the existence of small-amplitude generalized solitary-wave
solutions and modulated solitary-wave solutions to the steady gravity-capillary
Whitham equation with weak surface tension. Through the application of the
center manifold theorem, the nonlocal equation for the solitary wave profiles
is reduced to a four-dimensional system of ODEs inheriting reversibility. Along
particular parameter curves, relating directly to the classical
gravity-capillary water wave problem, the associated linear operator is seen to
undergo either a reversible $0^{2+}(i k_0)$ bifurcation or a reversible $(i
s)^2$ bifurcation. Through a normal form transformation, the reduced system of
ODEs along each relevant parameter curve is seen to be well approximated by a
truncated system retaining only second-order or third-order terms. These
truncated systems relate directly to systems obtained in the study of the full
gravity-capillary water wave equation and, as such, the existence of
generalized and modulated solitary waves for the truncated systems is
guaranteed by classical works, and they are readily seen to persist as
solutions of the gravity-capillary Whitham equation due to reversibility.
Consequently, this work illuminates further connections between the
gravity-capillary Whitham equation and the full two-dimensional
gravity-capillary water wave problem.
