Université de Strasbourg

Raphaël Côte

Biography - Raphaël Côte

Institute for Advanced Mathematical Research (IRMA), University of Strasbourg and CNRS, France

Raphaël Côte, USIAS Fellow 2022Raphaël Côte obtained his PhD in mathematics from the Cergy-Pontoise University (Paris, France) in 2006, under the supervision of Professor Frank Merle. He took up the position of instructor at the École normale supérieure (ENS) – of which he was a former student - from 2004 to 2007. He was then hired as research fellow at the French National Centre for Scientific Research (CNRS) and pursued his research at the Laurent Schwartz Center for Mathematics of the École polytechnique, from 2007 to 2016, where also was part-time professor from 2009 on. He was invited to the University of Princeton (USA) in 2003-2004, and to the University of Chicago (USA) in 2011-2012.

Since 2016, he is Professor at the University of Strasbourg: he held a research chair as part of its Excellence initiative (IdEx), and develops his research topics within the Institute for Advanced Mathematical Research (IRMA), as the head of the Modelisation and Control group. He was a recipient of the 2018 Guy Ourisson prize from the Gutenberg circle.

Raphaël Côte is interested in the understanding and the description of nonlinear phenomena originating from physics. An expert in the theoretical analysis of partial differential equations, he studies long time dynamics of solutions of evolution equations, in particular phenomena of wave propagation or concentration, and the role of dispersion: stability and interaction of solitons or self-similar solutions, formation of vortices or topological singularities, etc.

Project - Dynamics and control in micromagnetism

01/12/2022 – 30/11/2024

In recent years, remarkable progress has been made in the understanding of the dynamics of solutions to nonlinear dispersive partial differential equations (PDEs) like the Schrödinger equation, wave equations, or the (generalized) Korteweg-de Vries equation. One example is the soliton resolution for wave type equations, in particular for the damped nonlinear Klein-Gordon equation. In a paper written in collaboration with Yvan Martel and Xu Yuanit is shown that, on the one hand, any global solution (for positive times) decomposes into a finite sum of alternating solitons (stationary solutions), and that the distance between two consecutive solitons follows a universal law at leading order, and on the other hand, that all these configurations do indeed occur.

The idea at the heart of this project is to adapt these techniques to other settings or models, in particular in micro- and ferromagnetism, where many questions that currently remain open could be approached using techniques developed for dispersive PDEs.

One of the important models in micromagnetism is the Landau-Lifschitz-Gilbert equation, which models wave propagation and singularity formation in nano- or micro materials. Despite the fact that this model is neither Hamiltonian nor dispersive, like the equations mentioned above, the nonlinear objects that appear (domain walls and vortices) share many properties with solitons.

The aim of this project is threefold. The first aspect is to study dynamical properties of domain walls in the spirit of the techniques developed in the dispersive setting, specifically questions relating to interaction and collision. The second focuses on the effects of temperature on topological structures such as domain walls. The third is to consider control-related problems, and meets physics motivations: domain wall transition can be used to encode data, and one essential issue is to be able to modify this data in a precise, fast, reliable and economical way.

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