The Korteweg–de Vries equation at $H^{-1}$ regularity

Abstract

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 $H^{-1}$ bound 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.