  In this work, we consider the stability of solitons for the KdV equation
below the energy space, using spatially-exponentially-weighted norms. Using a
combination of the $I$-method and spectral analysis following Pego and
Weinstein, we are able to show that, in the exponentially weighted space, the
perturbation of a soliton decays exponentially for arbitrarily long times. The
finite time restriction is due to a lack of global control of the unweighted
perturbation.

Analysis of PDEs