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.
Categories: math.AP