Raphaël Côte

Raphaël Côte
Professor

Institut de Recherche Mathématique Avancée
UMR 7501, Université de Strasbourg
7 rue René-Descartes
67084 Strasbourg Cedex, France

Email :
Office :	i407
Phone :	+33 3 68 85 01 44

Research

I am a member of the Modelisation and Control (MOCO) group.
I take part in the Master of Mathematics focused on PDEs and machine learning offered during the academic year 2022/2023.
I am a USIAS fellow.

Research themes

Non-linear evolution PDEs.
Dynamics, stability, blow-up, asymptotic behaviour and classification of PDEs.
Wave equations, generalized Korteweg-de Vries equations, non-linear Schrödinger equations, dispersive equations.
Solitons, multi-solitons, breathers, self-similar solutions.
Micromagnetism, Landau-Lifshitz-Gilbert equation, Néel walls.

Preprints

Asymptotic stability of 2-domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction.
Raphaël Côte and Guillaume Ferrière.

Minimal time of magnetization switching in small ferromagnetic ellipsoidal samples.
Raphaël Côte, Clémentine Courtès, Guillaume Ferrière and Yannick Privat.

Domain walls for the Landau-Lifschitz equation with Dzyaloshinskii-Moriya interaction.
Raphaël Côte and Radu Ignat.

Perturbation at blow-up time of self-similar solutions for the modified Korteweg-de Vries equation.
Simão Correia and Raphaël Côte.

Publications

Improved uniqueness of multi-breathers of the modified Korteweg-de Vries equation.
Raphaël Côte and Alexander Semenov.
To appear in Vietnam Journal of Mathematics.

Concentration close to the cone for linear waves.
Raphaël Côte and Camille Laurent.
To appear in Revista Matemática Iberoamericana.

Spatial decay of multi-solitons of the generalized Korteweg-de Vries and nonlinear Schrödinger equations.
Raphaël Côte and Xavier Friederich.
To appear in Mathematische Annalen.

Asymptotics of solutions with a compactness property for the nonlinear damped Klein-Gordon equation.
Raphaël Côte and Xu Yuan.
Nonlinear Analysis. 218 (2022), 112768.

Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation.
Raphaël Côte, Yvan Martel and Xu Yuan.
Archive for Rational Mechanics and Analysis. 239 (2021) n^o 3, 1837-1874.

Description and classification of 2-solitary waves for nonlinear damped Klein-Gordon equations.
Raphaël Côte, Yvan Martel, Xu Yuan and Lifeng Zhao.
Communications in Mathematical Physics. 388 (2021), 1557-1601.

On smoothness and uniqueness of multi-solitons of the non linear Schrödinger equations.
Raphaël Côte and Xavier Friederich.
Communications in PDE. 46 (2021) n^o 12, 2325-2385.

Self-similar dynamics for the modified Korteweg-de Vries equation.
Simão Correia, Raphaël Côte and Luis Vega.
International Mathematics Research Notices. 2021 n^o 13, 9958-10013.

Polynomial growth of high Sobolev norms of solutions to the Zakharov-Kuznetsov equation.
Raphaël Côte and Frédéric Valet.
Communications on Pure and Applied Analysis. 20 (2021) n^o 3, 1039-1058.

Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation.
Simão Correia, Raphaël Côte and Luis Vega.
Journal de Mathématiques Pures et Appliquées. 137 (2020), 101-142.

Multi-travelling waves for the nonlinear Klein-Gordon equation.
Raphaël Côte and Yvan Martel.
Transactions of the American Mathematical Society. 370 (2018) n^o 10, 7461-7487.

Profiles for the radial focusing 4D energy-critical wave equation.
Raphaël Côte, Carlos E. Kenig, Andrew Lawrie and Wilhelm Schlag.
Communications in Mathematical Physics. 357 (2018) n^o 4, 943-1008.

Limiting motion for the parabolic Ginzburg-Landau equation with infinite energy data.
Raphaël Côte and Delphine Côte.
Communications in Mathematical Physics. 350 (2017) n^o 2, 507-568.

Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons.
Raphaël Côte, Claudio Muñoz, Didier Pilod and Gideon Simpson.
Archive for Rational Mechanics and Analysis. 220 (2016) n^o 2, 639-710.

On the soliton resolution for equivariant wave maps to the sphere.
Raphaël Côte.
Communications on Pure and Applied Mathematics. 68 (2015), 1946-2004.

Characterization of large energy solutions of the equivariant wave map problem: II.
Raphaël Côte, Carlos E. Kenig, Andrew Lawrie and Wilhelm Schlag.
American Journal of Mathematics. 137 (2015) n^o 1, 209-250.

Characterization of large energy solutions of the equivariant wave map problem: I.
Raphaël Côte, Carlos E. Kenig, Andrew Lawrie and Wilhelm Schlag.
American Journal of Mathematics. 137 (2015) n^o 1, 139-207.

A thin-film limit in the Landau-Lifshitz-Gilbert equation relevant for the formation of Néel walls.
Raphaël Côte, Radu Ignat and Evelyne Miot.
Journal of Fixed Point Theory and Applications. 15 (2014), The Haïm Brezis Festschrift, n^o 1, 241-272.

Multi-solitons for the nonlinear Klein-Gordon equation.
Raphaël Côte and Claudio Muñoz.
Forum of Mathematics, Sigma. 2 (2014), e15, 38 pp.

Energy partition for the linear wave equation.
Raphaël Côte, Carlos E. Kenig and Wilhelm Schlag.
Mathematische Annalen. 358 (2014), 573-607.

Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension.
Raphaël Côte and Hatem Zaag.
Communications on Pure and Applied Mathematics. 66 (2013) n^o 10, 1541-1581.

High speed excited multi-solitons in nonlinear Schrödinger equations.
Raphaël Côte and Stefan Le Coz.
Journal de Mathématiques Pures et Appliquées. 96 (2011), 135-166.

N-solitons for the L²-supercritical generalized Korteweg-de Vries and non-linear Schrödinger equations.
Raphaël Côte, Yvan Martel and Frank Merle.
Revista Matemática Iberoamericana. 27 (2011) n^o 1, 273-302.

Scattering below critical energy for the radial 4D Yang-Mills equation and for the 2D corotational wave map system.
Raphaël Côte, Carlos E. Kenig and Frank Merle.
Communications in Mathematical Physics. 284 (2008) n^o 1, 203-225.

Scaling-sharp dispersive estimates for the Korteweg-de Vries group.
Raphaël Côte and Luis Vega.
Comptes Rendus Mathématiques. Académie des Sciences. Paris. 346 (2008), 845-848.

Construction of solutions to the L²-critical KdV equation with a given asymptotic behaviour.
Raphaël Côte.
Duke Mathematical Journal. 138 (2007) n^o 3, 487-532.

Construction of solutions to the subcritical gKdV equations with a given asymptotic behaviour.
Raphaël Côte.
Journal of Functional Analysis. 241 (2006), 143-211.

Large data wave operator for the generalized Korteweg-de Vries equations.
Raphaël Côte.
Differential and Integral Equations. 19 (2006) n^o 2, 163-188.

Instability of non-constant harmonic maps for the (1+2)-dimensional equivariant wave map system.
Raphaël Côte.
International Mathematics Research Notices. 2005 n^o 57, 3525-3549.

Proceedings

Around self-similar solutions to the modified Korteweg-de Vries equation.
Raphaël Côte.
Oberwolfach reports. 16, no. 5, 331-334. (2019).

Self-similar solutions and critical spaces for the modified Korteweg-de Vries equation.
Raphaël Côte.
Séminaire Laurent-Schwartz. 2018-2019. Exp. 4, 15 pages. École Polytechnique, Palaiseau (2019).

Les solitons.
Raphaël Côte.
Gazette des Mathématiciens. 153 (2017), 59-63.

Asymptotic stability of solitons for the Zahkarov-Kuznetsov equation.
Raphaël Côte.
Oberwolfach reports. 13, no. 2, 1531-1534. (2016).

Soliton resolution for equivariant wave maps to the sphere.
Raphaël Côte.
Oberwolfach reports. 10, no. 3, 2331-2333. (2013).

Construction de solutions pour les équations de Korteweg-de Vries généralisées.
Raphaël Côte.
Séminaire: Équations aux Dérivées Partielles. 2006-2007. Exp. 3, 17 pages. École Polytechnique, Palaiseau (2007).

Solutions to the generalized Korteweg-de Vries equations with a prescribed asymptotic behavior.
Raphaël Côte.
Oberwolfach reports. 4, no. 4, 2637-2639. (2007).

Thesis

Construction et propriétés de solutions pour des équations dispersives focalisantes.
Raphaël Côte.
PhD thesis.

Solitons et dispersion.
Raphaël Côte.
Habilitation thesis.

Collaborators

I have the pleasure to collaborate with

I have (or had) the pleasure to supervise the PhD thesis of

Haiming Du, since 2022 (co-supervision at Peking University with Baoping Liu).
Guillaume Steimer, since 2021. (co-supervision with Emmanuel Franck, Laurent Navoret and Vincent Vigon)
Alexander Semenov. PhD defended in July 2022 at the University of Strasbourg.
Xavier Friederich. PhD defended in June 2021 at the University of Strasbourg.
Frédéric Valet. PhD defended in July 2020 at the University of Strasbourg.
Yakine Bahri. PhD defended in July 2016 at the École polytechnique (co-supervision with Philippe Gravejat).

and the post-doctorates of

Ludovic Godard-Cadillac (2022-2023).
Guillaume Ferrière (2021-2023).
Clémentine Courtès (2017-2018).
Simão Correia (2017-2018).

Teaching and Administration

In 2022/23, I teach the courses

Functional Analysis (MSc, 1st year),
Non linear evolution PDEs (MSc, 2nd year), with Clémentine Courtès.

Funding

USIAS Project Dynamics and control in micromagnetism, 2022-2024.
ANR Project MOSICOF (Modeling and Simulation of Complex Ferromagnetic Systems), 2021-2025.

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Last change: January 2023.