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We present some results obtained in collaboration with Simão Correia (University of Lisbon) and Luis Vega (University of Bilbao), regarding the understanding of self-similar solutions for the modified Korteweg-de Vries equation (mKdV). We obtain the description of self-similar solutions in Fourier space, and we also prove a local well-posedness result in a critical space where self-similar solutions live. As a consequence, we can study the flow of (mKdV) around self-similar solutions: in particular, we give an asymptotic description of small solutions as and construct solutions with a prescribed blow up behavior as .
@article{SLSEDP_2018-2019____A4_0,
author = {C\^ote, Rapha\"el},
title = {Self-similar solutions and critical spaces for {the~modified~Korteweg-de} {Vries} equation},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:4},
pages = {1--15},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2018-2019},
doi = {10.5802/slsedp.130},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.5802/slsedp.130/}
}
TY - JOUR AU - Côte, Raphaël TI - Self-similar solutions and critical spaces for the modified Korteweg-de Vries equation JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:4 PY - 2018-2019 SP - 1 EP - 15 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://geodesic.mathdoc.fr/articles/10.5802/slsedp.130/ DO - 10.5802/slsedp.130 LA - en ID - SLSEDP_2018-2019____A4_0 ER -
%0 Journal Article %A Côte, Raphaël %T Self-similar solutions and critical spaces for the modified Korteweg-de Vries equation %J Séminaire Laurent Schwartz — EDP et applications %Z talk:4 %D 2018-2019 %P 1-15 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U http://geodesic.mathdoc.fr/articles/10.5802/slsedp.130/ %R 10.5802/slsedp.130 %G en %F SLSEDP_2018-2019____A4_0
Côte, Raphaël. Self-similar solutions and critical spaces for the modified Korteweg-de Vries equation. Séminaire Laurent Schwartz — EDP et applications (2018-2019), Exposé no. 4, 15 p. doi : 10.5802/slsedp.130. http://geodesic.mathdoc.fr/articles/10.5802/slsedp.130/
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