  In this paper we will prove the existence of weak solutions to the
Korteweg-de Vries initial value problem on the real line with H^{-1} initial
data; moreover, we will study the problem of orbital and asymptotic H^{s}
stability of solitons for integers s\ge -1; finally, we will also prove new a
priori H^{-1} bounds for solutions to the Korteweg-de Vries equation. The paper
will utilise the Miura transformation to link the Korteweg-de Vries equation to
the modified Korteweg-de Vries equation.
