2025Activity reportProject-Team​‌PARADYSE

3.1 Time asymptotics: Stationary‌​‌ states, solitons, and stability​​ issues

The team investigates​​​‌ the existence of solitons‌ and their link with‌​‌ the global dynamical behavior​​ for non-local problems such​​​‌ as the Gross–Pitaevskii (GP)‌ equation which arises in‌​‌ models of dipolar gases.​​ These models, in general,​​​‌ also introduce non-zero boundary‌ conditions which constitute an‌​‌ additional theoretical and numerical​​ challenge. Numerous results are​​​‌ proved for local problems,‌ and numerical simulations allow‌​‌ to verify and illustrate​​ them, as well as​​​‌ making a link with‌ physics. However, most fundamental‌​‌ questions are still open​​ at the moment for​​​‌ non-local problems.

The non-linear‌ Schrödinger (NLS) equation finds‌​‌ applications in numerous fields​​ of physics. We concentrate,​​​‌ in a continued collaboration‌ with our colleagues from‌​‌ the physics department (​​PhLAM) at Université​​​‌ de Lille (U-Lille) in‌ the framework of the‌​‌ Laboratoire d'Excellence CEMPI, on​​ its applications in non-linear​​​‌ optics and cold atom‌ physics. Issues of orbital‌​‌ stability and modulational instability​​ are central here (see​​​‌ Section 4.1 below).

Another‌ typical example of problem‌​‌ that the team wishes​​ to address concerns the​​​‌ Landau–Lifshitz (LL) equation, which‌ describes the dynamics of‌​‌ the spin in ferromagnetic​​ materials. This equation is​​​‌ a fundamental model in‌ the magnetic recording industry‌​‌ 46 and solitons in​​ magnetic media are of​​​‌ particular interest as a‌ mechanism for data storage‌​‌ or information transfer 47​​. It is a​​​‌ quasilinear PDE involving a‌ function that takes values‌​‌ on the unit sphere​​ ${𝕊}^2$ of ${\mathrm{ℝ‌}}^3$. Using the‌ stereographic projection, it can‌​‌ be seen as a​​ quasilinear Schrödinger equation and​​​‌ the questions about the‌ solitons, their dynamics and‌​‌ potential blow-up of solutions​​​‌ evoked above are also​ relevant in this context.​‌ This equation is less​​ understood than the NLS​​​‌ equation: even the Cauchy​ theory is not completely​‌ understood 38, 45​​. In particular, the​​​‌ geometry of the target​ sphere imposes that the​‌ solution has a norm​​ equal to one everywhere,​​​‌ so in particular the​ boundary conditions cannot be​‌ zero, and, even in​​ dimension one, there are​​​‌ kink-type solitons having different​ limits at $\pm \mathrm{\infty ‌}$.

3.2 Derivation of​​ macroscopic laws from microscopic​​​‌ dynamics

The team investigates,​ from a microscopic viewpoint,​‌ the dynamical mechanism at​​ play in the phenomenon​​​‌ of relaxation towards thermal​ equilibrium for large systems​‌ of interacting particles. For​​ instance, a first step​​​‌ consists in giving a​ rigorous proof of the​‌ fact that a particle​​ repeatedly scattered by random​​​‌ obstacles through a Hamiltonian​ scattering process will eventually​‌ reach thermal equilibrium, thereby​​ completing previous works in​​​‌ this direction by the​ team. As a second​‌ step, models similar to​​ the ones considered classically​​​‌ will be defined and​ analyzed in the quantum​‌ mechanical setting, and more​​ particularly in the setting​​​‌ of quantum optics.

Another​ challenging problem is to​‌ understand the interaction of​​ large systems with the​​​‌ boundary, which is responsible​ for most energy exchanges​‌ (forcing and dissipation), even​​ though it is concentrated​​​‌ in very thin layers.​ The presence of boundary​‌ conditions to evolution equations​​ sometimes lacks understanding from​​​‌ a physical and mathematical​ point of view. In​‌ order to legitimate the​​ choice done at the​​​‌ macroscopic level of the​ mathematical definition of the​‌ boundary conditions, we investigate​​ systems of particles with​​​‌ different local interactions. We​ apply various techniques to​‌ understand how diffusive and​​ driven systems interact with​​​‌ the boundaries.

Finally, we​ aim at obtaining results​‌ on the macroscopic behavior​​ of large scale interacting​​​‌ particle systems subject to​ kinetic constraints. In particular,​‌ we study the behavior​​ in one and two​​​‌ dimensions of the Facilitated​ Exclusion Process (FEP), on​‌ which several results have​​ already been obtained. The​​​‌ latter is a very​ interesting prototype for kinetically​‌ constrained models because of​​ its unique mathematical features​​​‌ (explicit stationary states and​ absence of mobile cluster​‌ to locally shuffle the​​ configuration). There are very​​​‌ few mathematical results on​ the FEP, which was​‌ put forward by the​​ physics community as a​​​‌ toy model for phase​ separation.

Our goal is​‌ to develop collaboration at​​ the interface between probability​​​‌ and PDE theory, and​ use the rich PDE​‌ background of the team​​ to provide tools to​​​‌ be used on statistical​ physics problems put forward​‌ by the probability side​​ of the team.

3.3​​​‌ Numerical methods: analysis and​ simulations

The team addresses​‌ both questions of precision​​ and numerical cost of​​​‌ discrete schemes for the​ numerical integration of non-linear​‌ evolution PDEs, such as​​ the NLS equation. In​​​‌ particular, we aim at​ developing, studying and implementing​‌ numerical schemes with high-order​​ convergence rates that are​​​‌ more efficient for these​ problems. We also want​‌ to contribute to the​​ design and analysis of​​ schemes with appropriate qualitative​​​‌ properties. These properties may‌ as well be “asymptotic-preserving”‌​‌ properties, energy-preserving properties, or​​ convergence to an equilibrium.​​​‌ Other numerical goals of‌ the team include the‌​‌ numerical simulation of standing​​ waves of non-linear non-local​​​‌ GP equations. We also‌ keep on developing numerical‌​‌ methods to efficiently simulate​​ and illustrate theoretical results​​​‌ on instability, in particular‌ in the context of‌​‌ the modulational instability in​​ optical fibers, where we​​​‌ study the influence of‌ randomness in the physical‌​‌ parameters of the fibers.​​

The team also designs​​​‌ simulation methods to estimate‌ the accuracy of the‌​‌ physical description via microscopic​​ systems, by computing precisely​​​‌ the rate of convergence‌ as the system size‌​‌ goes to infinity. One​​ method under investigation is​​​‌ related to cloning algorithms,‌ which were introduced very‌​‌ recently and turn out​​ to be essential in​​​‌ molecular simulation.

4.1 Optical fibers‌​‌

Participants: Stephan De Bièvre​​, Guillaume Dujardin.​​​‌

In the propagation of‌ light in optical fibers,‌​‌ the combined effect of​​ non-linearity and group velocity​​​‌ dispersion (GVD) may lead‌ to the destabilization of‌​‌ the stationary states (plane​​ or continuous waves). This​​​‌ phenomenon, known under the‌ name of modulational instability‌​‌ (MI), consists in the​​ exponential growth of small​​​‌ harmonic perturbations of a‌ continuous wave. The study‌​‌ of MI has been​​ pioneered in the 60s​​​‌ in the context of‌ fluid mechanics, electromagnetic waves‌​‌ as well as in​​ plasmas, and it has​​​‌ been observed in non-linear‌ fiber optics in the‌​‌ 80s. In uniform fibers,​​ MI arises for anomalous​​​‌ (negative) GVD, but it‌ may also appear for‌​‌ normal GVD if polarization,​​ higher order modes or​​​‌ higher order dispersion are‌ considered. A different kind‌​‌ of MI related to​​ a parametric resonance mechanism​​​‌ emerges when the dispersion‌ or the non-linearity of‌​‌ the fiber are periodically​​ modulated.

As a follow-up​​​‌ of our work on‌ MI in periodically modulated‌​‌ optical fibers, we investigate​​ the effect of random​​​‌ modulations in the diameter‌ of the fiber on‌​‌ its dynamics. It is​​ expected on theoretical grounds​​​‌ that such random fluctuations‌ can lead to MI‌​‌ and this has already​​ been illustrated for some​​​‌ models of randomness. We‌ investigate precisely the conditions‌​‌ under which this phenomenon​​ can be strong enough​​​‌ to be experimentally verified.‌ For this purpose, we‌​‌ investigate different kinds of​​ random processes describing the​​​‌ modulations, taking into account‌ the manner in which‌​‌ such modulations can be​​ created experimentally by our​​​‌ partners of the fiber‌ facility of the PhLAM.‌​‌ This necessitates a careful​​ modeling of the fiber​​​‌ and a precise numerical‌ simulation of its behavior‌​‌ as well as a​​ theoretical analysis of the​​​‌ statistics of the fiber‌ dynamics.

4.2 Ferromagnetism

Participants:‌​‌ André De Laire Peirano​​, Guillaume Dujardin,​​​‌ Guillaume Ferrière.

The‌ Landau–Lifshitz (LL) equation describes‌​‌ the dynamics of the​​ spin in ferromagnetic materials.​​​‌ Depending on the properties‌ of the material, the‌​‌ LL equation can include​​ a dissipation term (the​​​‌ so-called Gilbert damping) and‌ different types of anisotropic‌​‌ terms. The LL equation​​​‌ belongs to a larger​ class of non-linear PDEs​‌ which are often referred​​ to as geometric PDEs,​​​‌ and some related models​ are the Schrödinger map​‌ equation and the harmonic​​ heat flow. We focus​​​‌ on the following aspects​ of the LL equation.​‌

• Solitons

  In the absence​​ of Gilbert damping, the​​​‌ LL equation is Hamiltonian.​ Moreover, it is integrable​‌ in the one-dimensional case​​ and explicit formulas for​​​‌ solitons can be given.​ In the easy-plane case,​‌ the orbital and asymptotic​​ stability of these solitons​​​‌ have been established. However,​ the stability in other​‌ cases, such as in​​ biaxial ferromagnets, remains an​​​‌ open problem. In higher​ dimensional cases, the existence​‌ of solitons is more​​ involved. In a previous​​​‌ work, a branch of​ semitopological solitons with different​‌ speeds has been obtained​​ numerically in planar ferromagnets.​​​‌ A rigorous proof of​ the existence of such​‌ solitons is established using​​ perturbation arguments, provided that​​​‌ the speed is small​ enough. However, the proof​‌ does not give information​​ about their stability. We​​​‌ would like to propose​ a variational approach to​‌ study the existence of​​ this branch of solitons,​​​‌ that would lead to​ the existence and stability​‌ of the whole branch​​ of ground-state solitons as​​​‌ predicted. We also investigate​ numerically the existence of​‌ other types of localized​​ solutions for the LL​​​‌ equation, such as excited​ states or vortices in​‌ rotation.

  On the other​​ hand, with the inclusion​​​‌ of the Gilbert damping,​ the Landau-Lifshitz-Gilbert (LLG) equation​‌ becomes (partially) dissipative. Interestingly,​​ in the one-dimensional case,​​​‌ the same solitons, referred​ to as domain walls​‌, emerge as significant​​ structures. Not only do​​​‌ they demonstrate asymptotic stability,​ even in the presence​‌ of a small magnetic​​ field (42),​​​‌ but they also serve​ as crucial building blocks​‌ for various stable configurations,​​ such as 2-domain wall​​​‌ structures (41).​ Numerical simulations further suggest​‌ that any general solution​​ should decompose over time​​​‌ into a superposition of​ domain walls, though this​‌ still presents an open​​ problem at the theoretical​​​‌ level. Exploring the scenario​ of a notched nanowire​‌ (40) reveals​​ yet another context where​​​‌ generalized domain walls manifest.​ They exhibit an even​‌ better asymptotic stability compared​​ to their non-notched counterparts,​​​‌ which may lead to​ applications in information storage.​‌

• Approximate models
  An important​​ physical conjecture is that​​​‌ the LL model is​ to a certain extent​‌ universal, so that the​​ non-linear Schrödinger and Sine-Gordon​​​‌ equations can be obtained​ as its various limit​‌ cases. In a previous​​ work, A. de Laire​​​‌ has proved a result​ in this direction and​‌ established an error estimate​​ in Sobolev norms, in​​​‌ any dimension. A next​ step is to produce​‌ numerical simulations that will​​ enlighten the situation and​​​‌ drive further developments in​ this direction.
• Self-similar behavior​‌
  Self-similar solutions have attracted​​ a lot of attention​​​‌ in the study of​ non-linear PDEs because they​‌ can provide some important​​ information about the dynamics​​​‌ of the equation. While​ self-similar expanders are related​‌ to non-uniqueness and long​​ time description of solutions,​​ self-similar shrinkers are related​​​‌ to a possible singularity‌ formation. However, there is‌​‌ not much known about​​ the self-similar solutions for​​​‌ the LL equation. A.‌ de Laire and S.‌​‌ Gutierrez (University of Birmingham)​​ have studied expander solutions​​​‌ and proved their existence‌ and stability in the‌​‌ presence of Gilbert damping.​​ We will investigate further​​​‌ results about these solutions,‌ as well as the‌​‌ existence and properties of​​ self-similar shrinkers.

4.3 Bose-Einstein​​​‌ condensates and nonlinear optics‌

Participants: Quentin Chauleur,‌​‌ André De Laire Peirano​​, Guillaume Dujardin,​​​‌ Guillaume Ferrière.

In‌ quantum physics and nonlinear‌​‌ optics, the Gross-Pitaevskii equation​​ with non-zero boundary conditions​​​‌ is employed to describe‌ the behavior of quantum‌​‌ fluids and Bose-Einstein condensates.​​ The primary challenges are​​​‌ to comprehend new realistic‌ physical effects, such as‌​‌ nonlocal interactions, quasilinear effects​​ and variations in the​​​‌ width of the domain.‌

In order to establish‌​‌ a rigorous understanding of​​ the dynamics of these​​​‌ models, the study of‌ particular solutions such as‌​‌ dark solitons, which play​​ a key role in​​​‌ the large-time behavior, is‌ a crucial first step.‌​‌ For instance, proving the​​ stability of dark solitons,​​​‌ based on various physical‌ considerations, implies that these‌​‌ structures are good candidates​​ to be controlled experimentally​​​‌ and to be considered‌ in new applications.

Although‌​‌ the properties of dark​​ solitons are well-known in​​​‌ classical models described by‌ the Gross-Pitaevskii equation, the‌​‌ situation becomes more intricate​​ when adding terms to​​​‌ model new realistic physical‌ effects. Each characteristic introduces‌​‌ a range of new​​ theoretical and numerical difficulties.​​​‌ This complexity emphasizes the‌ need for a careful‌​‌ and detailed examination to​​ enhance our understanding of​​​‌ these intricate systems.

4.4‌ Cold atoms

Participants: Quentin‌​‌ Chauleur, Stephan De​​ Bièvre, Guillaume Dujardin​​​‌.

The cold atoms‌ team of the PhLAM‌​‌ Laboratory is reputed for​​ having realized experimentally the​​​‌ so-called Quantum Kicked Rotor,‌ which provides a model‌​‌ for the phenomenon of​​ Anderson localization. The latter​​​‌ was predicted by Anderson‌ in 1958, who received‌​‌ in 1977 a Nobel​​ Prize for this work.​​​‌ Anderson localization is the‌ absence of diffusion of‌​‌ quantum mechanical wave functions​​ (and of waves in​​​‌ general) due to the‌ presence of randomness in‌​‌ the medium in which​​ they propagate. Its transposition​​​‌ to the Quantum Kicked‌ Rotor goes as follows:‌​‌ a freely moving quantum​​ particle periodically subjected to​​​‌ a “kick” will see‌ its energy saturate at‌​‌ long times. In this​​ sense, it “localizes” in​​​‌ momentum space since its‌ momenta do not grow‌​‌ indefinitely, as one would​​ expect on classical grounds.​​​‌ In its original form,‌ Anderson localization applies to‌​‌ non-interacting quantum particles and​​ the same is true​​​‌ for the saturation effect‌ observed in the Quantum‌​‌ Kicked Rotor.

The challenge​​ is now to understand​​​‌ the effects of interactions‌ between the atoms on‌​‌ the localization phenomenon. Transposing​​ this problem to the​​​‌ Quantum Kicked Rotor, this‌ means describing the interactions‌​‌ between the particles with​​ a Gross–Pitaevskii equation, which​​​‌ is a NLS equation‌ with a local (typically‌​‌ cubic) non-linearity. So the​​​‌ particle's wave function evolves​ between kicks following the​‌ Gross–Pitaevskii equation and not​​ the linear Schrödinger equation,​​​‌ as is the case​ in the Quantum Kicked​‌ Rotor. Preliminary studies for​​ the Anderson model have​​​‌ concluded that in that​ case the localization phenomenon​‌ gives way to a​​ slow subdiffusive growth of​​​‌ the particle's kinetic energy.​ A similar phenomenon is​‌ expected in the non-linear​​ Quantum Kicked Rotor, but​​​‌ a precise understanding of​ the dynamical mechanisms at​‌ work, of the time​​ scale at which the​​​‌ subdiffusive growth will occur​ and of the subdiffusive​‌ growth exponent is lacking.​​ It is crucial to​​​‌ design and calibrate the​ experimental setup intended to​‌ observe the phenomenon. The​​ analysis of these questions​​​‌ poses considerable theoretical and​ numerical challenges due to​‌ the difficulties involved in​​ understanding and simulating the​​​‌ long term dynamics of​ the non-linear system. A​‌ collaboration of the team​​ members with the PhLAM​​​‌ cold atoms group is​ currently under way.

4.5​‌ Modelling shallow water dynamics​​

Participants: André De Laire​​​‌ Peirano, Olivier Goubet​.

The understanding of​‌ the propagation of waves​​ in shallow water is​​​‌ of importance for the​ modelling of tsunamis and​‌ other rogue waves. This​​ requires a better understanding​​​‌ of dispersive shallow water​ systems as ABCD systems,​‌ that are related to​​ the classical Boussines systems,​​​‌ and classifying particular travelling​ waves solutions for these​‌ systems. To deal with​​ systems is at forefront​​​‌ of research. Analogous questions​ for single equations as​‌ KdV equations are well-documented.​​

This project includes collaboration​​​‌ with researchers in Chile​ : C. Muñoz (Universidad​‌ de Chile), M. E.​​ Martinez (University of Chile)​​​‌ and F. Poblete (Austral​ University of Chile). The​‌ applications for tsunamis is​​ of interest for people​​​‌ in Chile.

4.6 Qualitative​ and quantitative properties of​‌ numerical methods

Participants: Quentin​​ Chauleur, Guillaume Dujardin​​​‌.

4.7 Mathematical modeling for​​ ecology

Participant: Olivier Goubet​​​‌.

This application domain‌ is at the interface‌​‌ of mathematical modeling and​​ numerics. Its object of​​​‌ study is a set‌ of concrete problems in‌​‌ ecology. The landscape of​​ the south of the​​​‌ Hauts-de-France region is made‌ of agricultural land, encompassing‌​‌ forest patches and ecological​​ corridors such as hedges.​​​‌ The issues are:

• the‌ study of the invasive‌​‌ dynamics and the control​​ of a population of​​​‌ beetles which damages the‌ oaks and beeches of‌​‌ our forests;
• the study​​ of native protected species​​​‌ (the purple wireworm and‌ the pike-plum) which find‌​‌ refuge in certain forest​​ species.

Running numerics on​​​‌ models co-constructed with ecologists‌ is also at the‌​‌ heart of the project.​​ In our model, the​​​‌ timescales of animals and‌ plants compare: the life‌​‌ cycle of a tree​​ is around one year,​​​‌ while (for animals) we‌ consider mainly insects whose‌​‌ life cycle is also​​ of one year, even​​​‌ for the propagation of‌ insects. For instance, beetle‌​‌ larvae spend a few​​ years in the earth​​​‌ before moving. As a‌ by-product, the mathematical model‌​‌ may tackle other major​​ issues such as the​​​‌ interplay between heterogeneity, diversity‌ and invasibility.

The models‌​‌ use Markov chains at​​ a mesoscopic scale and​​​‌ evolution advection-diffusion equations at‌ a macroscopic scale.

5.1 Footprint of research​​​‌ activities

The team is‌ committed to addressing environmental‌​‌ challenges and try to​​ minimize its carbon footprint.​​​‌ Whenever possible, train travel‌ is prioritized for conferences‌​‌ and research visits. Several​​ team members have adopted​​​‌ a no-flight policy for‌ ecological reasons.

5.2 Impact‌​‌ of research results

Our​​ work as applied mathematicians​​​‌ is often interdisciplinary. We‌ work with other scientists‌​‌ on cold atoms problems,​​ quantum information theory, forests​​​‌ modelling, etc. Our societal‌ relevance mostly comes from‌​‌ our inputs in advances​​ in these applied research​​​‌ directions.

6.1 Latest software developments​​

7.1 On the​‌ stationary solution of the​​ Landau-Lifshitz-Gilbert equation on a​​​‌ nanowire with constant external​ magnetic field

Participant: Guillaume​‌ Ferrière.

In the​​ preprint 21, Guillaume​​​‌ Ferriere examines the Landau-Lifshitz-Gilbert​ (LLG) equation governing the​‌ magnetization dynamics in an​​ infinite ferromagnetic nanowire with​​​‌ easy-axis anisotropy along the​ $e_1$ direction and​‌ subjected to a constant​​ external magnetic field ${\mathrm{h‌}}_0{e}_1$.​ Under specific conditions on​‌ $h_0$, the​​ study establishes the existence​​​‌ of stationary solutions with​ identical asymptotic behavior at​‌ infinity, their uniqueness up​​ to the symmetries of​​​‌ the LLG equation, and​ the instability of their​‌ orbits under the LLG​​ flow. These findings provide​​​‌ new insights into the​ behavior of solutions to​‌ the LLG equation, complemented​​ by numerical simulations that​​​‌ explore the stability of​ 2-domain wall structures and​‌ the interactions between domain​​ walls.

7.2 Existence and​​​‌ Uniqueness of Domain Walls​ for Notched Ferromagnetic Nanowires​‌

Participant: Guillaume Ferrière.​​

In the paper 17​​​‌, in collaboration with​ R. Côte (University of​‌ Strasbourg), C. Courtès (University​​ of Strasbourg), Guillaume Ferriere​​​‌ , L. Godard-Cadillac (University​ of Bordeaux), and Y.​‌ Privat (University of Lorraine,​​ Nancy) explore the existence​​​‌ and properties of domain​ walls in a model​‌ of notched ferromagnetic nanowires.​​ They employ variational methods​​​‌ and critical point theory​ to investigate the energy​‌ functional describing the system.​​

The authors first establish​​​‌ the equivalence of the​ critical points of this​‌ functional and the critical​​ points of another, more​​​‌ suitable functional through lifting.​ The existence of a​‌ minium is then achieved​​ under the assumption that​​​‌ the residual cross-section area​ function $s$ is strictly​‌ below 1 in a​​ bounded interval and is​​​‌ equal to 1 outside​ this interval. They then​‌ demonstrate the uniqueness of​​ the critical point under​​​‌ the proper constraints on​ the limits at $\pm ‌\infty$ by leveraging a​​ Mountain-Pass argument. The uniqueness​​​‌ requires stronger monotonicity assumptions,​ mainly that $s$ is​‌ unimodal, all the more​​ as it is expected​​​‌ that non-uniqueness should hold​ in the case of​‌ many notches.

The identified​​ critical point corresponds to​​​‌ a domain wall structure,​ i.e. a transition from​‌ $-e_1$ to​​ $e_1$. The​​​‌ authors also prove that​ the transition is mainly​‌ performed inside the notch.​​ Furthermore, the study analyzes​​ the asymptotic behavior of​​​‌ the solution, showing that‌ the magnetization decays to‌​‌ a uniform state at​​ infinity. In the special​​​‌ case of a symmetric‌ notch, additional insights are‌​‌ obtained using rearrangement techniques.​​

7.3 Minimal time of​​​‌ magnetization switching in small‌ ferromagnetic ellipsoidal samples.

Participant:‌​‌ Guillaume Ferrière.

In​​ the article 18,​​​‌ R. Côte (Université de‌ Strasbourg), C. Courtès (Université‌​‌ de Strasbourg), Guillaume Ferriere​​ and Y. Privat (Université​​​‌ de Lorraine & Institut‌ Universitaire de France) studied‌​‌ the process of the​​ switching of the magnetization​​​‌ of a small ferromagnetic‌ ellipsoidal sample, assumed to‌​‌ be constant, by an​​ external magnetic field. They​​​‌ first proved that the‌ latter must be strong‌​‌ enough in order to​​ be able to switch​​​‌ the magnetization. When this‌ condition is fullfiled, they‌​‌ found an explicit expression​​ for the minimal time​​​‌ of switching in cases‌ where the sample satisfies‌​‌ some rotational symmetry property.​​ Some numerical simulations complete​​​‌ the study.

7.4 The‌ Spear and the Ring:‌​‌ Emergent Structures in Magnetic​​ Colloidal Suspensions

Participant: Guillaume​​​‌ Ferrière.

In the‌ preprint 32, R.‌​‌ Côte (Université de Strasbourg),​​ C. Courtès (Université de​​​‌ Strasbourg), Guillaume Ferriere ,‌ Ludovic Godard-Cadillac (Université de‌​‌ Bordeaux) and Y. Privat​​ (Université de Lorraine &​​​‌ Institut Universitaire de France)‌ studied from a mathematical‌​‌ point of view the​​ nanoparticle model of a​​​‌ magnetic colloid presented by‌ G. Klughertz, in which‌​‌ small ferromagnetic particles immerged​​ in a fluid interact​​​‌ with each other through‌ standard dipole-dipole magnetic interaction‌​‌ and soft sphere model.​​ More specifically, they analyzed​​​‌ two specific structures: the‌ spear (chain of aligned‌​‌ particles) and the ring​​ (particles placed on a​​​‌ circle with $N$-fold‌ rotationnal symmetry). Under some‌​‌ technical assumptions on the​​ parameters of the interactions,​​​‌ they proved the existence‌ and the uniqueness of‌​‌ these structures, along with​​ bounds and sharp asymptotics​​​‌ (as the number of‌ particles tends to infinity)‌​‌ of the distance between​​ neighboring particles.

7.5 On​​​‌ the propagation of high‌ regularity for the logarithmic‌​‌ Schrödinger equation

Participants: Quentin​​ Chauleur, Guillaume Ferrière​​​‌.

In the prepint‌ 30, Quentin Chauleur‌​‌ and Guillaume Ferriere managed​​ to understand the effect​​​‌ of cancellation points in‌ the instantaneous loss of‌​‌ regularity of the logarithmic​​ nonlinear Schrödinger equation, which​​​‌ was numerically observed and‌ had been an open‌​‌ question for many years​​ on the theoretical level.​​​‌

7.6 On the dependence‌ of the nonlinear Schrodinger‌​‌ flow upon the power​​ of the nonlinearity

Participants:​​​‌ Quentin Chauleur, Guillaume‌ Ferrière.

In collaboration‌​‌ with R. Carles (Université​​ de Rennes), Quentin Chauleur​​​‌ and Guillaume Ferriere proved‌ in 28 some precise‌​‌ convergence rates for the​​ limit of classical nonlinear​​​‌ Schrödinger models with power‌ nonlinearities to the logarithmic‌​‌ nonlinear Schrödinger equation equation​​ in the subcritical regime​​​‌ and in large time.‌

7.7 On the ground‌​‌ state of the nonlinear​​ Schrödinger equation: asymptotic behavior​​​‌ at the endpoint powers‌

Participants: Quentin Chauleur,‌​‌ Guillaume Ferrière.

In​​ collaboration with R. Carles​​​‌ (Université de Rennes) and‌ D. Pelinovsky (McMaster University),‌​‌ Quentin Chauleur and Guillaume​​​‌ Ferriere studied in 29​ both limits of the​‌ stationary nonlinear Schrödinger equation​​ when the power of​​​‌ the nonlinearity tends to​ zero or to the​‌ energy-critical exponent, up to​​ some rescaling to infer​​​‌ non-trivial limits. They proved​ strong convergence results for​‌ both cases, and provide​​ detailed numerical insights.

7.8​​​‌ Gray and black solitons​ of nonlocal Gross-Pitaevskii equations​‌

Participant: André De Laire​​ Peirano.

In the​​​‌ paper 24, Andre​ De Laire Peirano and​‌ S. López-Martínez (Autonomous University​​ of Madrid) continue the​​​‌ investigation started in previous​ works concerning the qualitative​‌ aspects of dark solitons​​ of one-dimensional Gross-Pitaevskii equations​​​‌ with general nonlocal interactions.​ Under general conditions on​‌ the potential interaction term,​​ they provide uniform bounds,​​​‌ demonstrate the existence of​ symmetric solitons, and identify​‌ conditions under which monotonicity​​ is lost. Additionally, they​​​‌ present new properties of​ black solitons. Moreover, they​‌ establish the nonlocal-to-local convergence,​​ i.e. the convergence of​​​‌ the soliton of the​ nonlocal model toward the​‌ explicit dark solitons of​​ the local Gross-Pitaevskii equation.​​​‌

7.9 Exotic traveling waves​ for a quasilinear Schrödinger​‌ equation with nonzero background​​

Participants: André De Laire​​​‌ Peirano, Erwan Le​ Quiniou.

Andre De​‌ Laire Peirano and Erwan​​ Le Quiniou have studied​​​‌ a quasilinear Schrödinger equation​ with nonzero conditions at​‌ infinity in dimension one.​​ This quasilinear model corresponds​​​‌ to a weakly nonlocal​ approximation of the nonlocal​‌ Gross–Pitaevskii equation, and can​​ also be derived by​​​‌ considering the effects of​ surface tension in superfluids.​‌ When the quasilinear term​​ is neglected, the resulting​​​‌ equation is the classical​ Gross–Pitaevskii equation, which possesses​‌ a well-known stable branch​​ of subsonic traveling waves​​​‌ solution, given by dark​ solitons.

In the paper​‌ 23, they investigate​​ how the quasilinear term​​​‌ affects the traveling-waves solutions.​ They provide a complete​‌ classification of finite energy​​ traveling waves of the​​​‌ equation, in terms of​ the two parameters: the​‌ speed and the strength​​ of the quasilinear term.​​​‌ This classification leads to​ the existence of dark​‌ and antidark solitons, as​​ well as more exotic​​​‌ localized solutions like dark​ cuspons, compactons, and composite​‌ waves, even for supersonic​​ speeds. Depending on the​​​‌ parameters, these types of​ solutions can coexist, showing​‌ that finite energy solutions​​ are not unique. Furthermore,​​​‌ they prove that some​ of these dark solitons​‌ can be obtained as​​ minimizers of the energy,​​​‌ at fixed momentum, and​ that they are orbitally​‌ stable.

7.10 Traveling waves​​ for a quasilinear Schrödinger​​​‌ equation

Participant: Erwan Le​ Quiniou.

In the​‌ paper 26, Erwan​​ Le Quiniou studies a​​​‌ quasilinear Schrödinger equation with​ nonzero conditions at infinity.​‌ In the previous work​​ 23 with Andre De​​​‌ Laire Peirano , he​ obtained a continuous branch​‌ of traveling waves, given​​ by dark solitons indexed​​​‌ by their speed. Neglecting​ the quasilinear term, one​‌ recovers the Gross–Pitaevskii equation,​​ for which the branch​​​‌ of dark solitons is​ stable. It is known​‌ that the Vakhitov–Kolokolov (VK)​​ stability criterion or momentum​​​‌ of stability criterion holds​ for general semilinear equations​‌ with nonvanishing conditions at​​ infinity. In the quasilinear​​ case, E. Le Quiniou​​​‌ proves that the VK‌ stability criterion still applies‌​‌ and he deduces that​​ the branch of dark​​​‌ solitons is stable for‌ weak quasilinear interactions. For‌​‌ stronger quasilinear interactions, a​​ cusp appears in the​​​‌ energy-momentum diagram, implying the‌ stability of fast waves‌​‌ and the instability of​​ slow waves.

7.11 Travelling​​​‌ waves for the Gross–Pitaevskii‌ equation on the strip‌​‌

Participant: André De Laire​​ Peirano.

In one​​​‌ space dimension, the Gross-Pitaevskii‌ equation possesses a family‌​‌ of finite energy travelling​​ waves, called dark solitons.​​​‌ These solitons extend trivially‌ to the strip given‌​‌ by the product space​​ $ℝ\times {𝕋}_{\mathrm{L‌}}$, where $L>‌0$ and ${𝕋}_{\mathrm{L‌‌}}$ is the torus ${\mathrm{𝕋}}_L=ℝ/‌L\mathbb{Z}.$ In‌ this two-dimensional context, the‌​‌ dark solitons are called​​ planar (or line) dark​​​‌ solitons. However, it is‌ well-known in the physics‌​‌ literature that these planar​​ solitons can be unstable​​​‌ due to the tendency‌ to develop distortions in‌​‌ their transverse profile. In​​ addition, experimental observations have​​​‌ shown that the dynamics‌ of planar dark solitons‌​‌ are stable when they​​ are sufficiently confined in​​​‌ the transverse direction $\mathrm{L‌}$, but unstable otherwise.‌​‌ In the latter case,​​ the creation of vortices​​​‌ can occur.

In the‌ article 22, Andre‌​‌ De Laire Peirano ,​​ P. Gravejat (CY Cergy​​​‌ Paris University) and D.‌ Smets (Sorbonne University) provide‌​‌ a rigorous framework for​​ studying this kind of​​​‌ phenomenon. Precisely, they prove‌ the existence of nonconstant‌​‌ finite energy travelling wave​​ solutions to the Gross-Pitaevskii​​​‌ equation on the strip‌ $ℝ\times {𝕋}_{\mathrm{L‌‌}}$, obtained as minimizers​​ of the energy at​​​‌ fixed momentum. Moreover, by‌ studying the associated variational‌​‌ problem, they deduce that​​ these minimizers are exactly​​​‌ the planar dark solitons‌ when $L$ is less‌​‌ than a critical value,​​ and that they are​​​‌ genuinely two-dimensional solutions otherwise.‌ In particular, planar solitons‌​‌ do not minimize the​​ energy in the presence​​​‌ of a large transverse‌ direction. The proof of‌​‌ the existence of minimizers​​ is based on the​​​‌ compactness of minimizing sequences,‌ relying on a new‌​‌ symmetrization argument that is​​ well-suited to the periodic​​​‌ setting.

7.12 Standing wave‌ for two-dimensional Schrödinger equations‌​‌ with discontinuous dispersion

Participants:​​ Olivier Goubet.

In​​​‌ collaboration with B. Alouini‌ (University of Monastir) and‌​‌ I. Manoubi (Université of​​ Gabès) 15, Olivier​​​‌ Goubet has studied the‌ existence and stability of‌​‌ standing wave for an​​ evolution nonlinear Schrödinger equation​​​‌ with discontinuous dispersion, the‌ discontinuity being supported by‌​‌ a straight line. Both​​ pure power nonlinearities and​​​‌ logarithmic nonlinearities are considered.‌ The discontinuity destroys the‌​‌ invariance by space translation​​ for the equation. The​​​‌ main result is that‌ when restricted to a‌​‌ suitable subspace that contains​​ the standing waves, these​​​‌ waves are orbitally stable‌ in the $H^{\mathrm{1‌‌}}$ subcritical regime in the​​ pure power case or​​​‌ in the logarithmic case,‌ and strongly unstable in‌​‌ the critical or supercritical​​ case.

7.13 Dynamics of​​​‌ generalized abcd Boussinesq solitary‌ waves under a slowly‌​‌ variable bottom

Participants: André​​​‌ De Laire Peirano,​ Olivier Goubet.

In​‌ collaboration with the associated​​ team PANDA, the team​​​‌ investigated the existence of​ generalized solitary waves and​‌ the corresponding collision problem​​ in the physically relevant​​​‌ variable bottom regime, for​ the $ab\mathrm{c‌}d$-Boussinesq system. In​​ 33, the authors​​​‌ provide a detailed description​ of weak long-range interactions​‌ and the evolution of​​ the traveling wave without​​​‌ its destruction. They establish​ this result by constructing​‌ a new approximate solution​​ that captures the interaction​​​‌ between the solitary wave​ and a slowly varying​‌ bottom.

7.14 Kirkwood-Dirac distributions​​

Participants: Stephan De Bièvre​​​‌, Christopher Langrenez,​ Mateo Spriet.

The​‌ Kirkwood-Dirac (KD) quasiprobability distribution​​ can describe any quantum​​​‌ state with respect to​ the eigenbases of two​‌ observables A and B.​​ KD distributions behave similarly​​​‌ to classical joint probability​ distributions but can assume​‌ negative and nonreal values.​​ In 43, Stephan​​​‌ De Bièvre provided an​ in-depth study of the​‌ notion of completely incompatible​​ observables that he recently​​​‌ introduced and of its​ links to the support​‌ uncertainty and to the​​ Kirkwood-Dirac nonpositivity of pure​​​‌ quantum states. The latter​ notion has recently been​‌ proven central to a​​ number of issues in​​​‌ quantum information theory and​ quantum metrology. In this​‌ last context, it was​​ shown that a quantum​​​‌ advantage requires the use​ of Kirkwood-Dirac nonclassical states.​‌

Several papers have been​​ published by members of​​​‌ PARADYSE on this subject​ in the last year.​‌ They are mentioned in​​ the following sections.

7.15 Strong convergence​​​‌ for the discrete nonlinear‌ Klein-Gordon equation

Participant: Quentin‌​‌ Chauleur.

In 16​​, Quentin Chauleur extends​​​‌ the analysis of nonlinear‌ dispersive equations, such as‌​‌ the nonlinear Klein-Gordon equation,​​ on an infinite lattice​​​‌ $h{\mathbb{Z}}^d$ as‌ the lattice spacing $\mathrm{h‌‌}\to 0$ approaches the​​ continuum limit. This work​​​‌ builds upon the framework‌ established in previous works,‌​‌ employing bilinear estimates of​​ the Shannon interpolation combined​​​‌ with controls on the‌ growth of discrete Sobolev‌​‌ norms of the solutions.​​ Additionally, Quentin Chauleur also​​​‌ provides some perspectives on‌ uniform dispersive estimates for‌​‌ nonlinear waves on lattices.​​

7.16 Vortex nucleation in​​​‌ 2D rotating Bose–Einstein condensates‌

Participants: Quentin Chauleur,‌​‌ Guillaume Dujardin.

In​​ 20, Guillaume Dujardin​​​‌ , I. Lacroix-Violet (University‌ of Lorraine, Nancy) and‌​‌ A. Nahas (University of​​ Lille) introduce a new​​​‌ numerical method for the‌ minimization under constraints of‌​‌ a discrete energy modeling​​ multicomponents rotating Bose-Einstein condensates​​​‌ in the regime of‌ strong confinement and with‌​‌ rotation. Moreover, they consider​​ both segregation and coexistence​​​‌ regimes between the components.‌ It is well known‌​‌ that, depending on the​​ regime, the minimizers may​​​‌ display different structures, sometimes‌ with vorticity (from singly‌​‌ quantized vortices, to vortex​​ sheets and giant holes).​​​‌ In order to study‌ numerically the structures of‌​‌ the minimizers, the authors​​ of 20 introduce a​​​‌ numerical algorithm for the‌ computation of the indices‌​‌ of the vortices, as​​ well as an algorithm​​​‌ for the computation of‌ the indices of vortex‌​‌ sheets. Several computations are​​ carried out, to illustrate​​​‌ the efficiency of the‌ method, to cover different‌​‌ physical cases, to validate​​ recent theoretical results as​​​‌ well as to support‌ conjectures. Moreover, the new‌​‌ methods is compared with​​ an alternative method from​​​‌ the literature. This work‌ was part of A.‌​‌ Nahas' PhD thesis, co-advised​​ by I. Lacroix-Violet (University​​​‌ of Lorraine, Nancy) and‌ Guillaume Dujardin .

7.17‌​‌ High order uniform in​​ time schemes for weakly​​​‌ nonlinear Schrödinger equation and‌ wave turbulence

Participant: Quentin‌​‌ Chauleur.

In the​​ preprint 31, Quentin​​​‌ Chauleur and A. Mouzard‌ (Université de Nanterre) has‌​‌ developed some uniform in​​ time high order schemes​​​‌ for the discretization of‌ NLS equations with a‌​‌ small (weak) nonlinearity. The​​​‌ motivation of such work​ lies in the theory​‌ of wave turbulence, which​​ describes the nonlinear interactions​​​‌ of waves outside thermal​ equilibrium by a statistical​‌ approach, in analogy with​​ Boltzmann kinetic theory of​​​‌ gases. This analysis aims​ for instance at understanding​‌ the behavior of waves​​ propagating at the surface​​​‌ of the ocean, with​ the coexistence of waves​‌ of various wavelengths propagating​​ in many directions. The​​​‌ scheme developed was further​ used in order to​‌ study such physical behaviors.​​

8.1 International initiatives

8.2​‌ National initiatives

8.3​​​‌ Regional initiatives

9.1 Promoting​‌ scientific activities

9.2 Teaching -​‌ Supervision - Juries -​​ Educational and pedagogical outreach​​​‌

Participants: Quentin Chauleur,​ Stephan De Bièvre,​‌ André De Laire Peirano​​, Guillaume Dujardin,​​​‌ Olivier Goubet, Guillaume​ Ferrière.

10.1 Major‌ publications

• 1 articleR.‌​‌Ragaa Ahmed, C.​​Cedric Bernardin, P.​​​‌Patricia Gonçalves and M.‌Marielle Simon. A‌​‌ Microscopic Derivation of Coupled​​ SPDE's with a KPZ​​​‌ Flavor.Annales de‌ l'Institut Henri Poincaré58‌​‌22022HALDOI​​
• 2 articleC.Cédric​​​‌ Bernardin, P.Patricia‌ Gonçalves, M.Milton‌​‌ Jara and M.Marielle​​ Simon. Interpolation process​​​‌ between standard diffusion and‌ fractional diffusion.Annales‌​‌ de l'Institut Henri Poincaré​​ (B) Probabilités et Statistiques​​​‌5432018,‌ 1731 - 1757HAL‌​‌DOI
• 3 articleC.​​Christophe Besse, S.​​​‌Stephane Descombes, G.‌Guillaume Dujardin and I.‌​‌Ingrid Lacroix-Violet. Energy​​ preserving methods for nonlinear​​​‌ Schrödinger equations.IMA‌ Journal of Numerical Analysis‌​‌411January 2021​​, 618–653HALDOI​​​‌
• 4 articleO.Oriane‌ Blondel, C.Clément‌​‌ Erignoux and M.Marielle​​ Simon. Stefan problem​​​‌ for a non-ergodic facilitated‌ exclusion process.Probability‌​‌ and Mathematical Physics2​​12021HALDOI​​​‌
• 5 articleQ.Quentin‌ Chauleur. Growth of‌​‌ Sobolev norms and strong​​​‌ convergence for the discrete​ nonlinear Schrödinger equation.​‌Nonlinear Analysis: Theory, Methods​​ and Applications242May​​​‌ 2024, 113517HAL​DOI
• 6 articleR.​‌Raphaël Côte and G.​​Guillaume Ferriere. Asymptotic​​​‌ stability of 2-domain walls​ for the Landau-Lifshitz-Gilbert equation​‌ in a nanowire with​​ Dzyaloshinskii-Moriya interaction.International​​​‌ Mathematics Research Notices2024​4February 2024,​‌ 3551-3600HALDOI
• 7​​ articleS.Stephan De​​​‌ Bièvre. Complete Incompatibility,​ Support Uncertainty, and Kirkwood-Dirac​‌ Nonclassicality.Physical Review​​ Letters2021HALDOI​​​‌
• 8 articleS.Serge​ Dumont, O.Olivier​‌ Goubet and Y.Youcef​​ Mammeri. Decay of​​​‌ solutions to one dimensional​ nonlinear Schrödinger equations with​‌ white noise dispersion.​​Discrete and Continuous Dynamical​​​‌ Systems - Series S​1482021,​‌ 2877-2891HALDOI
• 9​​ articleC.Clément Erignoux​​​‌. Hydrodynamic limit for​ an active exclusion process​‌.Mémoires de la​​ Société Mathématique de France​​​‌169May 2021HAL​DOI
• 10 articleC.​‌Célia Griffet, M.​​Matthieu Arnhem, S.​​​‌Stephan De Bièvre and​ N.Nicolas Cerf.​‌ Interferometric measurement of the​​ quadrature coherence scale using​​​‌ two replicas of a​ quantum optical state.​‌Physical Review A108​​2August 2023,​​​‌ 023730HALDOI
• 11​ articleA.Anaelle Hertz​‌ and S.Stephan de​​ Bièvre. Quadrature coherence​​​‌ scale driven fast decoherence​ of bosonic quantum field​‌ states.Physical Review​​ LettersMarch 2020HAL​​​‌DOI
• 12 articleA.​André de Laire,​‌ P.Philippe Gravejat and​​ D.Didier Smets.​​​‌ Construction of minimizing travelling​ waves for the Gross-Pitaevskii​‌ equation on $$.Tunisian​​ Journal of Mathematics6​​​‌12024, 157-188​In press. HALDOI​‌
• 13 articleA.André​​ de Laire and P.​​​‌Philippe Gravejat. The​ cubic Schrödinger regime of​‌ the Landau-Lifshitz equation with​​ a strong easy-axis anisotropy​​​‌.Revista Matemática Iberoamericana​3712021,​‌ 95-128HALDOI
• 14​​ articleA.André de​​​‌ Laire and S.Salvador​ López-Martínez. Existence and​‌ decay of traveling waves​​ for the nonlocal Gross-Pitaevskii​​​‌ equation.Communications in​ Partial Differential Equations47​‌92022, 1732-1794​​HALDOI

10.2 Publications​​​‌ of the year

10.3 Cited​‌ publications

• 38 articleI.​​Ioan Bejenaru, A.​​​‌ D.Alexandru D Ionescu​, C. E.Carlos​‌ E Kenig and D.​​Daniel Tataru. Global​​​‌ Schrödinger maps in dimensions​ $d2$: small​‌ data in the critical​​ Sobolev spaces.Annals​​​‌ of Mathematics2011,​ 1443--1506back to text​‌
• 39 articleC.Christophe​​ Besse, S.Stephane​​​‌ Descombes, G.Guillaume​ Dujardin and I.Ingrid​‌ Lacroix-Violet. Energy preserving​​ methods for nonlinear Schrödinger​​​‌ equations.IMA Journal​ of Numerical Analysis41​‌1January 2021,​​ 618--653HALDOIback​​​‌ to text
• 40 unpublished​G.Gilles Carbou and​‌ D.David Sanchez.​​ Stabilization of walls in​​​‌ notched ferromagnetic nanowires.​June 2018, Preprint​‌HALback to text​​
• 41 articleR.Raphaël​​​‌ Côte and G.Guillaume​ Ferriere. Asymptotic Stability​‌ of 2-Domain Walls for​​ the Landau–Lifshitz–Gilbert Equation in​​​‌ a Nanowire With Dzyaloshinskii–Moriya​ Interaction.International Mathematics​‌ Research Notices11 2023​​, rnad249URL: https://doi.org/10.1093/imrn/rnad249​​​‌DOIback to text​
• 42 articleR.Raphaël​‌ Côte and R.Radu​​ Ignat. Asymptotic stability​​​‌ of precessing domain walls​ for the Landau-Lifshitz-Gilbert equation​‌ in a nanowire with​​ Dzyaloshinskii-Moriya interaction.Comm.​​​‌ Math. Phys.4013​2023, 2901--2957URL:​‌ https://doi.org/10.1007/s00220-023-04714-9DOIback to​​ text
• 43 articleS.​​​‌S. De Bièvre.​ Relating incompatibility, noncommutativity, uncertainty,​‌ and Kirkwood--Dirac nonclassicality.​​Journal of Mathematical Physics​​642February 2023​​​‌, URL: http://dx.doi.org/10.1063/5.0110267DOI‌back to text
• 44‌​‌ articleG.Guillaume Dujardin​​, F.Frédéric Hérau​​​‌ and P.Pauline Lafitte-Godillon‌. Coercivity, hypocoercivity, exponential‌​‌ time decay and simulations​​ for discrete Fokker- Planck​​​‌ equations.Numerische Mathematik‌144https://arxiv.org/abs/1802.02173v12020HAL‌​‌DOIback to text​​
• 45 articleR. L.​​​‌Robert L. Jerrard and‌ D.Didier Smets.‌​‌ On Schrödinger maps from​​ $T^1$ to ${\mathrm{S‌}}^2$.Ann. Sci.‌ ENS452012,‌​‌ 637-680back to text​​
• 46 bookD.Dan​​​‌ Wei. Micromagnetics and‌ Recording Materials.http://dx.doi.org/10.1007/978-3-642-28577-6‌​‌Springer--Verlag Berlin Heidelberg2012​​, URL: https://doi.org/10.1007/978-3-642-28577-6back​​​‌ to text
• 47 article‌S.Shilei Zhang,‌​‌ A. A.Alexander A.​​ Baker, S.Stavros​​​‌ Komineas and T.Thorsten‌ Hesjedal. Topological computation‌​‌ based on direct magnetic​​ logic communication.Scientific​​​‌ Reports52015,‌ URL: http://dx.doi.org/10.1038/srep15773back to‌​‌ text