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
L
2
-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
L
2
-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
Simão Correia
Clémentine Courtès
Delphine Côte
Xavier Friederich
Radu Ignat
Carlos E. Kenig
Camille Laurent
Andrew Lawrie
Stefan Le Coz
Yvan Martel
Frank Merle
Evelyne Miot
Claudio Muñoz
Didier Pilod
Yannick Privat
Wilhelm Schlag
Alexander Semenov
Gideon Simpson
Frédéric Valet
Luis Vega
Xu Yuan
Hatem Zaag
Lifeng Zhao
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.