The Korteweg–de Vries equation at H -1 regularity
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, pp. 1071-1098.

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-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.

DOI: 10.1016/j.anihpc.2014.05.004
Keywords: Korteweg–de Vries equation, Stability of solitons, Miura map
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     title = {The {Korteweg{\textendash}de} {Vries} equation at $ {H}^{-1}$ regularity},
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Buckmaster, Tristan; Koch, Herbert. The Korteweg–de Vries equation at $ {H}^{-1}$ regularity. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, pp. 1071-1098. doi : 10.1016/j.anihpc.2014.05.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.05.004/

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