Raphaël Côte’s research while affiliated with French National Centre for Scientific Research and other places

This is the final report of the ANR project 14-CE25-0009-01 entitled "Mathematical Analysis of Topological Singularities in some physical problems" (MAToS) that was developed by the authors between January 2015-December 2019. The central theme of this project lied in the area of nonlinear analysis (nonlinear partial differential equations and calculus of variations). We focused on the structure and dynamics of topological singularities arising in some variational physical models driven by the Landau-Lifshitz equation (in micromagnetics) and the Gross-Pitaevskii equation (in superconductivity, Bose-Einstein condensation, nonlinear optics). These included vortex singularities, traveling waves and domain walls in magnetic thin films. These structures are observed experimentally and in numerical simulations and play an important role in the dynamics of the corresponding physical systems. We made significant progress in the mathematical analysis of these structures (both at the stationary and dynamical level) that gives more insight into the physical phenomena.

In this article, we investigate a simple model of notched ferromagnetic nanowires using tools from calculus of variations and critical point theory. Specifically, we focus on the case of a single unimodal notch and establish the existence and uniqueness of the critical point of the energy. This is achieved through a lifting argument, which reduces the problem to a generalized Sturm-Liouville equation. Uniqueness is demonstrated via a Mountain-Pass argument, where the assumption of two distinct critical points leads to a contradiction. Additionally, we show that the solution corresponds to a system of magnetic spins characterized by a single domain wall localized in the vicinity of the notch. We further analyze the asymptotic decay of the solution at infinity and explore the symmetric case using rearrangement techniques.

Non radiative solutions of the energy critical non linear wave equation are global solutions u that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$: $\lim_{t \to \pm \infty} \| \nabla_{t,x} u(t) \|_{L^2(|x| \ge |t|+R)} = 0,$ for some $R \gt 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see \cite{DKM:23} and the references therein. We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at 0 is given by non radiative solutions to the linear equation (described in \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.

We prove a first stability result of self-similar blow-up for the modified Korteweg–de Vries equation on the line. More precisely, given a self-similar solution and a sufficiently small regular profile, there is a unique global solution which behaves at t=0t=0 as the sum of the self-similar solution and the smooth perturbation.

We consider a ferromagnetic nanowire, with an energy functional E with easy-axis in the direction $e_{1}$, and which takes into account the Dzyaloshinskii–Moriya interaction. We consider configurations of the magnetization that are perturbations of two well-separated domain wall and study their evolution under the Landau–Lifshitz–Gilbert flow associated to E. Our main result is that, if the two walls have opposite speed, these configurations are asymptotically stable, up to gauges intrinsic to the invariances of the energy E. Our analysis builds on the framework developed in [4], taking advantage that it is amenable to space localisation.

We consider multi-breathers of (mKdV). In Semenov (2022), a smooth multi-breather was constructed, and proved to be unique in two cases: first, in the class of super-polynomial convergence to the profile (in the spirit of (Commun. Partial Differ. Equ. 46, 2325–2385, 2021)), and second, under the assumption that all speeds of the breathers involved are positive (without rate of convergence). The goal of this short note is to improve the second result: we show that uniqueness still holds if at most one velocity is negative or zero.

We consider a ferromagnetic nanowire and we focus on an asymptotic regime where the Dzyaloshinskii-Moriya interaction is taken into account. First we prove a dimension reduction result via $\Gamma$-convergence that determines a limit functional E defined for maps $m:\mathbb {R}\rightarrow \mathbb {S}^2$ in the direction $e_1$ of the nanowire. The energy functional E is invariant under translations in $e_1$ and rotations about the axis $e_1$. We fully classify the critical points of finite energy E when a transition between $-e_1$ and $e_1$ is imposed; these transition layers are called (static) domain walls. The evolution of a domain wall by the Landau–Lifshitz–Gilbert equation associated to E under the effect of an applied magnetic field $h(t)e_1$ depending on the time variable t gives rise to the so-called precessing domain wall. Our main result proves the asymptotic stability of precessing domain walls for small h in $L^\infty ([0, +\infty ))$ and small $H^1(\mathbb {R})$ perturbations of the static domain wall, up to a gauge which is intrinsic to invariances of the functional E.