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

DOI : https://doi.org/10.1016/j.anihpc.2014.05.004
Keywords: Korteweg–de Vries equation, Stability of solitons, Miura map
@article{AIHPC_2015__32_5_1071_0,
author = {Buckmaster, Tristan and Koch, Herbert},
title = {The Korteweg--de Vries equation at ${H}^{-1}$ regularity},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {32},
number = {5},
year = {2015},
pages = {1071-1098},
doi = {10.1016/j.anihpc.2014.05.004},
zbl = {1331.35300},
mrnumber = {3400442},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2015__32_5_1071_0}
}

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://www.numdam.org/item/AIHPC_2015__32_5_1071_0/

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