  We prove an asymptotic stability result for the water wave equations
linearized around small solitary waves. The equations we consider govern
irrotational flow of a fluid with constant density bounded below by a rigid
horizontal bottom and above by a free surface under the influence of gravity
neglecting surface tension. For sufficiently small amplitude waves, with
waveform well-approximated by the well-known sech-squared shape of the KdV
soliton, solutions of the linearized equations decay at an exponential rate in
an energy norm with exponential weight translated with the wave profile. This
holds for all solutions with no component in (i.e., symplectically orthogonal
to) the two-dimensional neutral-mode space arising from infinitesimal
translational and wave-speed variation of solitary waves. We also obtain
spectral stability in an unweighted energy norm.
