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 the head of the Modelisation and Control (MOCO) group.

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

• On smoothness and uniqueness of multi-solitons of the non linear Schrödinger equations.
  Raphaël Côte and Xavier Friederich.
  


• Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation.
  Raphaël Côte, Yvan Martel and Xu Yuan.
  


• Description and classification of 2-solitary waves for nonlinear damped Klein-Gordon equations.
  Raphaël Côte, Yvan Martel, Xu Yuan and Lifeng Zhao.
  

Publications

• Polynomial growth of high Sobolev norms of solutions to the Zakharov-Kuznetsov equation.
  Raphaël Côte and Frédéric Valet.
  To appear in Communications on Pure and Applied Analysis.


• 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.


• Self-similar dynamics for the modified Korteweg-de Vries equation.
  Simão Correia, Raphaël Côte and Luis Vega.
  International Mathematics Research Notices. 2020 n^o 01.


• 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 Mathematica 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. (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.

Editorial board

• I am a member of the editorial board of the Cours spécialisés of the French Mathematical Society (SMF).

Collaborators

• Simão Correia
• Delphine Côte
• Xavier Friederich
• Radu Ignat
• Carlos E. Kenig
• Andrew Lawrie
• Stefan Le Coz
• Yvan Martel
• Frank Merle
• Evelyne Miot
• Claudio Muñoz
• Didier Pilod
• Wilhelm Schlag
• Gideon Simpson
• Frédéric Valet
• Luis Vega
• Xu Yuan
• Hatem Zaag
• Lifeng Zhao

• Alexander Semenov, since 2018.
• Xavier Friederich, since 2018.
• Frédéric Valet. PhD defended in July 2020 at the Universtiy of Strasbourg.
• Yakine Bahri. PhD defended in July 2016 at the École polytechnique (co-supervision with Philippe Gravejat).

• Clémentine Courtès (2017-2018).
• Simão Correia (2017-2018).

Teaching and Administration

• Integration and Probabilities for the Agrégation (MSc, 2nd year),
• Functional Analysis (MSc, 1st year),
• Calculus (BSc, 1st year).

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Last change: October 2020.