The Korteweg–de Vries equation at H−1H−1 regularity

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−1H−1 initial data; moreover, we will study the problem of orbital and asymptotic HsHs stability of solitons for integers s≥−1s≥−1; finally, we will also prove new a priori H−1H−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.