  We study asymptotic stability of solitary wave solutions in the
one-dimensional Benney-Luke equation, a formally valid approximation for
describing two-way water wave propagation. For this equation, as for the full
water wave problem, the classic variational method for proving orbital
stability of solitary waves fails dramatically due to the fact that the second
variation of the energy-momentum functional is infinitely indefinite. We
establish nonlinear stability in energy norm under the spectral stability
hypothesis that the linearization admits no non-zero eigenvalues of
non-negative real part. We also verify this hypothesis for waves of small
energy.
