Self-similar solutions and critical spaces for the modified Korteweg-de Vries equation
Raphaël Côte1
1 Université de Strasbourg CNRS, IRMA UMR 7501 F-67000 Strasbourg, France
Séminaire Laurent Schwartz — EDP et applications (2018-2019), Exposé no. 4, 15 p.

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 t→+∞ and construct solutions with a prescribed blow up behavior as t→0.

Publié le :
DOI : 10.5802/slsedp.130
Raphaël Côte 1

1 Université de Strasbourg CNRS, IRMA UMR 7501 F-67000 Strasbourg, France 
@article{SLSEDP_2018-2019____A4_0,
     author = {Rapha\"el C\^ote},
     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 = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.130/}
}
TY  - JOUR
AU  - Raphaël Côte
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  - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.130/
DO  - 10.5802/slsedp.130
LA  - en
ID  - SLSEDP_2018-2019____A4_0
ER  - 
%0 Journal Article
%A Raphaël Côte
%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 https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.130/
%R 10.5802/slsedp.130
%G en
%F SLSEDP_2018-2019____A4_0
Raphaël Côte. 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. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.130/

[1] V. Banica and L. Vega. Evolution of polygonal lines by the binormal flow. , 2018. | arXiv

[2] Fernando Bernal-Vilchis and Pavel I. Naumkin. Self-similar asymptotics for solutions to the intermediate long wave equation. J. Evol. Equ., 2019. To appear. | DOI | MR | Zbl

[3] Simão Correia, Raphaël Côte, and Luis Vega. Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation. , 2018. | arXiv | DOI | MR

[4] Simão Correia, Raphaël Côte, and Luis Vega. Self-similar dynamics for the modified Korteweg-de Vries equation. , 2019. | arXiv | DOI

[5] P. A. Deift and X. Zhou. Asymptotics for the Painlevé II equation. Comm. Pure Appl. Math., 48(3):277–337, 1995. | DOI | Zbl

[6] A. S. Fokas and M. J. Ablowitz. On the initial value problem of the second Painlevé transcendent. Comm. Math. Phys., 91(3):381–403, 1983. | DOI | Zbl

[7] Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov. Painlevé transcendents. The Riemann-Hilbert approach, volume 128 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2006. | Zbl

[8] Pierre Germain, Fabio Pusateri, and Frédéric Rousset. Asymptotic stability of solitons for mKdV. Adv. Math., 299:272–330, 2016. | DOI | MR | Zbl

[9] R.E. Goldstein and D.M. Petrich. Soliton’s, Euler’s equations, and vortex patch dynamics. Phys. Rev. Lett., 69(4):555–558, 1992. | DOI | MR | Zbl

[10] Axel Grünrock and Luis Vega. Local well-posedness for the modified KdV equation in almost critical H r s ^-spaces. Trans. Amer. Math. Soc., 361:5681–5694, 2009. | DOI | MR | Zbl

[11] Benjamin Harrop-Griffiths. Long time behavior of solutions to the mKdV. Comm. Partial Differential Equations, 41(2):282–317, 2016. | DOI | MR | Zbl

[12] S. P. Hastings and J. B. McLeod. A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal., 73(1):31–51, 1980. | DOI | Zbl

[13] Nakao Hayashi and Pavel Naumkin. On the modified Korteweg-de Vries equation. Math. Phys. Anal. Geom., 4(3):197–227, 2001. | DOI | Zbl

[14] Nakao Hayashi and Pavel I. Naumkin. Large time behavior of solutions for the modified Korteweg-de Vries equation. Internat. Math. Res. Notices, (8):395–418, 1999. | DOI | Zbl

[15] Tosio Kato. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. In Studies in applied mathematics, volume 8 of Adv. Math. Suppl. Stud., pages 93–128. Academic Press, New York, 1983. | Zbl

[16] Carlos E. Kenig, Gustavo Ponce, and Luis Vega. Well-posedness and scattering result for the generalized Korteweg-De Vries equation via contraction principle. Comm. Pure Appl. Math., 46:527–620, 1993. | DOI | MR | Zbl

[17] Carlos E. Kenig, Gustavo Ponce, and Luis Vega. On the concentration of blow up solutions for the generalized KdV equation critical in L 2 . In Nonlinear wave equations (Providence, RI, 1998), volume 263 of Contemp. Math., pages 131–156. Amer. Math. Soc., Providence, RI, 2000. | DOI | Zbl

[18] Yvan Martel and Frank Merle. A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math. Pures Appl. (9), 79:339–425, 2000. | DOI | MR | Zbl

[19] G. Perelman and L. Vega. Self-similar planar curves related to modified Korteweg-de Vries equation. J. Differential Equations, 235(1):56–73, 2007. | DOI | MR | Zbl

Cité par Sources :