  In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of
the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries
(mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and
breather solutions, they demonstrate the lack of local well-posedness for these
equations below their respective endpoint regularities. In this paper, we study
the defocusing analogues of these equations, namely defocusing NLS, defocusing
mKdV, and real KdV, all in one spatial dimension, for which suitable soliton
and breather solutions are unavailable. We construct for each of these
equations classes of modified scattering solutions, which exist globally in
time, and are asymptotic to solutions of the corresponding linear equations up
to explicit phase shifts. These solutions are used to demonstrate lack of local
well-posedness in certain Sobolev spaces,in the sense that the dependence of
solutions upon initial data fails to be uniformly continuous. In particular, we
show that the mKdV flow is not uniformly continuous in the $L^2$ topology,
despite the existence of global weak solutions at this regularity.
  Finally, we investigate the KdV equation at the endpoint regularity
$H^{-3/4}$, and construct solutions for both the real and complex KdV
equations. The construction provides a nontrivial time interval $[-T,T]$ and a
locally Lipschitz continuous map taking the initial data in $H^{-3/4}$ to a
distributional solution $u \in C^0 ([-T,T]; $H^{-3/4})$ which is uniquely
defined for all smooth data. The proof uses a generalized Miura transform to
transfer the existing endpoint regularity theory for mKdV to KdV.
