2024Activity reportProject-TeamPARADYSE

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 53 and solitons in magnetic media are of particular interest as a mechanism for data storage or information transfer 54. It is a quasilinear PDE involving a function that takes values on the unit sphere ${𝕊}^2$ of ${ℝ}^3$. 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 44, 52. 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 $\pm \infty$.

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 boundaries, 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, 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 the 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 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 properties. 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

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. 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 the 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.

This application domain involves in particular S. De Bièvre and G. Dujardin.

4.2 Ferromagnetism

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 (49), but they also serve as crucial building blocks for various stable configurations, such as 2-domain wall structures (48). 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 (47) 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.

This application domain involves in particular A. de Laire, G. Dujardin and G. Ferriere.

4.3 Bose-Einstein condensates and nonlinear optics

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.

This application domain involves in particular A. de Laire, G. Dujardin, G. Ferriere and Q. Chauleur.

4.4 Cold atoms

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.

This application domain involves in particular S. De Bièvre, G. Dujardin and Q. Chauleur.

4.5 Modelling shallow water dynamics

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.

A. de Laire and O. Goubet are involved in these topics, together 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

4.7 Modeling of the liquid-solid transition and interface propagation

Analogously to the so-called Kinetically Constrained Models (KCM) that have served as toy models for glassy transitions, stochastic particle systems on a lattice can be used as toy models for a variety of physical phenomena. Among them, the kinetically constrained lattice gases (KCLG) are models in which particles jump randomly on a lattice, but are only allowed to jump if a local constraint is satisfied by the system.

Because of the hard constraint, the typical local behavior of KCLGs will differ significantly depending on the value of local conserved fields (e.g. particle density), because the constraint will either be typically satisfied, in which case the system is locally diffusive (liquid phase), or not, in which case the system quickly freezes out (solid phase).

Such a toy model for liquid-solid transition is investigated by the former member of the team C. Erignoux and his co-authors in 4 and 46. The focus of these articles is the so-called facilitated exclusion process, which is a terminology coined by physicists for a specific KCLG, in which particles can only jump on an empty neighbor if another neighboring site is occupied. They derive the macroscopic behavior of the model, and show that in dimension 1 the hydrodynamic limit displays a phase separated behavior where the liquid phase progressively invades the solid phase.

Both from a physical and mathematical point of view, much remains to be done regarding these challenging models: in particular, they present significant mathematical difficulties because of the way the local physical constraints put on the system distort the equilibrium and steady-states of the model. For this reason, A. Roget is currently working with C. Erignoux (DRACULA Project-team, Institut Camille Jordan, Université Lyon 1), M. Simon (Institut Camille Jordan, Université Lyon 1) and A. Shapira (MAP5, Paris) to generate numerical results on generalizations of the facilitated exclusion process, in order to shine some light on the microscopic and macroscopic behavior of these difficult models.

This application domain involves in particular A. Roget.

4.8 Mathematical modeling for ecology

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 one year. For animals we consider mainly insects whose life cycle is also of one year, even for the propagation of insects. 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.

This application domain involves O. Goubet. Interactions with PARADYSE members concerned with particle models and hydrodynamic limits are planned.

5.1 New software

6.1 Asymptotic stability of 2-domain walls for the Landau-Lifshitz-Gilbert equation in a nanowire with Dzyaloshinskii-Moriya interaction

The article 20 extends the study of magnetization dynamics in an infinite ferromagnetic nanowire, where the evolution is governed by the Landau-Lifshitz-Gilbert (LLG) equation. The energy functional considered includes an easy-axis anisotropy along the direction $e_1$ and incorporates the Dzyaloshinskii-Moriya interaction. In a previous work 49, R. Côte (University of Strasbourg) and R. Ignat (Toulouse Mathematics Institute) analyzed a specific structure, called domain wall, which connects $-e_1$ at $-\infty$ to $e_1$ at $+\infty$. They proved its uniqueness and asymptotic stability, up to translations and rotations around $e_1$, under a small external magnetic field. Their approach relied on energy properties near the domain wall and its evolution via the LLG flow.

In this paper 20, R. Côte (University of Strasbourg) and G. Ferriere investigate solutions which look like 2-domain walls, i.e. configurations with two well-separated transitions, each resembling a domain wall. They establish that, under specific conditions on the external magnetic field such that it drives the transitions further apart, these structures are also asymptotically stable, up to translations and rotations around $e_1$ for each transition. The proof employs energy methods analogous to those used in multi-soliton results for dispersive equations. By localizing the energy around each transition, they recover similar stability properties, with additional negligible terms.

6.2 Logarithmic Gross-Pitaevskii equation

The logarithmic nonlinearity in the context of Schrödinger equations has recently regained interest in various domains of physics. For instance, this model may generalize the Gross-Pitaevskii equation, used in the case of two-body interaction, to the case of three-body interaction. R. Carles (University of Rennes) and G. Ferriere study this equation, named the logarithmic Gross-Pitaevskii equation (or logGP), on the whole space ${ℝ}^d$ in 16. As the first mathematical study of this equation in this framework, they focus on its global wellposedness in the energy space, which turns out to correspond to the energy space for the standard Gross-Pitaevskii equation with a cubic nonlinearity in small dimensions, and on the characterization of solitary and traveling waves in the one-dimensional case. This works opens the door to further studies on this equation, especially on its asymptotic and long-time dynamics : multidimensional solitary and traveling waves and their orbital stability, scattering, multi-solitons, convergence towards other models...

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

In the preprint 36, G. 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 $h_0{e}_1$. 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.

6.4 Existence and Uniqueness of Domain Walls for Notched Ferromagnetic Nanowires

In the preprint 34, R. Côte (University of Strasbourg), C. Courtès (University of Strasbourg), G. 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.

6.5 Numerical computation of dark solitons of a nonlocal nonlinear Schrödinger equation

The existence and decay properties of dark solitons for a large class of nonlinear nonlocal Gross-Pitaevskii equations with nonzero boundary conditions in dimension one was established recently by A. de Laire and S. López-Martínez (Autonomous University of Madrid) in 14. Mathematically, these solitons correspond to minimizers of the energy at fixed momentum and are orbitally stable. In the paper 25, A. de Laire, G. Dujardin and S. López-Martínez (Autonomous University of Madrid) provide a numerical method to compute approximations of such solitons for these types of equations, and give actual numerical experiments for several types of physically relevant nonlocal potentials. These simulations allow them to obtain a variety of dark solitons, and to comment on their shapes in terms of the parameters of the nonlocal potential. In particular, they suggest that, given the dispersion relation, the speed of sound and the Landau speed are important values to understand the properties of these dark solitons. They also allow them to test the necessity of some sufficient conditions in the theoretical result proving existence of the dark solitons.

6.6 Gray and black solitons of nonlocal Gross-Pitaevskii equations

In the preprint 39, A. de Laire and S. López-Martínez (Autonomous University of Madrid) continue the investigation started in 14, 25, 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.

6.7 Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background

A. de Laire and E. 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 preprint 38, 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.

6.8 Traveling waves for a quasilinear Schrödinger equation

In the paper 42, E. Le Quiniou studies a quasilinear Schrödinger equation with nonzero conditions at infinity. In the previous work 38 with A. de Laire, 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.

6.9 Travelling waves for the Gross–Pitaevskii equation on the strip

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 $ℝ\times {𝕋}_L$, where $L>0$ and ${𝕋}_L$ is the torus ${𝕋}_L=ℝ/L\mathbb{Z}.$ 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 $L$, but unstable otherwise. In the latter case, the creation of vortices can occur.

In the articles 27 and 26, A. de Laire, 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 $ℝ\times {𝕋}_L$, 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.

6.10 Standing wave for two-dimensional Schrödinger equations with discontinuous dispersion

In collaboration with B. Alouini (University of Monastir) and I. Manoubi (Université of Gabès) 30, O. 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 $H^1$ subcritical regime in the pure power case or in the logarithmic case, and strongly unstable in the critical or supercritical case.

6.11 The logarithmic Schrödinger equation with spatial white noise on the full space

The logarithmic Schrödinger appears as a fundamental model in quantum gravity and nuclear physics, and adding a white noise potential can model strong media disorder. In 19, Q. Chauleur and A. Mouzard (University of Nanterre) prove the existence and uniqueness of solutions to the stochastic logarithmic Schrödinger. The proof relies on a particular exponential transform which have proved being useful in several contexts, in particular in models arising from quantum field theory.

6.12 Non-Linear Problems in Interacting Particle Systems

In his thesis 29, G. Nahum presents three studies on interacting particle systems where a form of nonlinearity emerges.

In the context of describing non-equilibrium steady states (NESS), he formulates an approach based on matrix products to characterize the steady state of a process defined on a lattice composed of sites numbered from 1 to $N$. This process evolves as a symmetric simple exclusion process (SSEP) on the intermediate sites, while being coupled with reaction-diffusion processes acting on pairs of boundary sites $\{1,2\}$ and $\{N-1,N\}$. Each pair of sites can adopt one of four possible states, resulting in 12 possible transitions. He derives a set of constraints ensuring the consistency of the underlying quadratic algebra, which are linked to reservoir correlations. He also presents a representation of the objects involved in this formulation and provides examples of transition rates that satisfy these constraints.

In the second study, he focuses on generalizing the porous media model (PMM), which is associated with a hydrodynamic equation where the diffusion rate depends on the density raised to a certain power. He extends this model by introducing a universal exclusion family parameterized by an exponent, which can take real values. This generalization allows the representation of the transition from the slow diffusion regime to the fast diffusion regime. He successfully addresses the case where the exponent lies in a specific interval, deriving the porous media equation for certain values of the exponent and the fast diffusion equation for others.

He further generalizes the PMM by constructing a diffusion coefficient that depends on a density multiplied by a function of the complementary density, parameterized by two exponents. The generalized model inherits certain theoretical properties of the original PMM, such as the presence of mobile clusters and blocked configurations. The construction of this model is delicate to preserve the gradient property of the PMM.

Finally, he extends this generalization to a long-range dynamics while maintaining the gradient property. This long-range dynamics is simple and can be applied to any exclusion process.

6.13 Kirkwood-Dirac distributions

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 50, S. 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.

6.14 Rigorous results on approach to thermal equilibrium, entanglement, and nonclassicality of an optical quantum field mode scattering from the elements of a non-equilibrium quantum reservoir

Rigorous derivations of the approach of individual elements of large isolated systems to a state of thermal equilibrium, starting from arbitrary initial states, are exceedingly rare. This is particularly true for quantum mechanical systems. In 21, S. De Bièvre and his collaborators demonstrate how, through a mechanism of repeated scattering, an approach to equilibrium of this type actually occurs in a specific quantum optics system.

6.15 Growth of Sobolev norms and strong convergence for the discrete nonlinear Schrödinger equation

As it is known, the nonlinear Schrödinger stands as a prime model in order to describe the propagation of waves in nonlinear optics or the dynamics of a superfluid in Bose-Einstein condensates. Its discretization in space stands as a first step in order to perform reliable and efficient numerical simulations. Q. Chauleur studies the convergence of the discrete nonlinear Schrödinger equation on a lattice $h{\mathbb{Z}}^d$ towards the continuous model as the step size of the lattice $h$ tends to zero in 18. The proof of the convergence relies on uniform dispersive estimates in order to control the growth of the discrete Sobolev norms of the solution, as well as bilinear estimates of the Shannon interpolation.

6.16 Strong convergence for the discrete nonlinear Klein-Gordon equation

In 31, Q. 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 $h\to 0$ approaches the continuum limit. This work builds upon the framework established in 18, employing bilinear estimates of the Shannon interpolation combined with controls on the growth of discrete Sobolev norms of the solutions. Additionally, Q. Chauleur also provides some perspectives on uniform dispersive estimates for nonlinear waves on lattices.

6.17 Linearly implicit high-order numerical methods for evolution problems

G. Dujardin and I. Lacroix-Violet (University of Lorraine, Nancy) introduced a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs, in the research direction detailed in Section 3.3. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. A specific analysis of Runge–Kutta collocation methods for this purpose was carried out by G. Dujardin and I. Lacroix-Violet (University of Lorraine, Nancy) 22. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods proved to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so 23.

6.18 Vortex nucleation in 2D rotating Bose–Einstein condensates

In 24, G. 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 24 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 G. Dujardin.

6.19 Finite volumes for the Gross-Pitaevskii equation

In 33, Q. Chauleur studies the approximation of the Gross-Pitaevskii equation with a time-dependent potential using a Voronoi finite-volume scheme. The time integration is handled via an explicit splitting scheme, while the spatial integration employs a two-point flux approximation finite-volume method. This work serves as the theoretical counterpart to the companion paper 32, providing a numerical analysis of the new scheme designed to explore the dynamics of Bose-Einstein condensates in various geometries.

6.20 Numerical study of the Gross-Pitaevskii equation on a two-dimensional ring and vortex nucleation

In 32, Q. Chauleur and G. Dujardin, in collaboration with physicists R. Chicireanu, J.-C. Garreau, and A. Rançon from the PhLAM laboratory at University of Lille, numerically investigate the dynamics of a Bose-Einstein condensate of cold potassium atoms confined by a ring potential with a Gaussian profile. By introducing a rotating sinusoidal perturbation, they demonstrate the nucleation of quantum vortices under specific dynamical regimes. The numerical simulations are carried out using Strang splitting for time integration and a two-point flux approximation finite-volume scheme applied to a carefully constructed admissible triangulation. Additionally, they develop numerical algorithms for vortex tracking tailored to the finite-volume framework.

6.21 Discrete quantum harmonic oscillator and Kravchuk transform

In 17, Q. Chauleur and E. Faou (University of Rennes) consider a particular discretization of the harmonic oscillator which admits an orthogonal basis of eigenfunctions called Kravchuk functions possessing appealing properties from the numerical point of view. We analytically prove the almost second-order convergence of these discrete functions towards Hermite functions, uniformly for large numbers of modes. We then describe an efficient way to simulate these eigenfunctions and the corresponding transformation. We finally show some numerical experiments corroborating our different results.

6.22 Numerical analysis of a semi-implicit Euler scheme for the Keller-Segel model

In a collaboration with X. Huang (School of Mathematical Science, Fujian, China) and J. Shen (School of Mathematical Science, Eastern Institute of Technology, Zhejiang, China) 37, O. Goubet has performed the numerical analysis of a discrete time scheme for a Keller-Segel model in dimension 2. This semi-implicit scheme preserves important features of the original equation as positivity of and diffusion.

7.1 International initiatives

7.2 International research visitors

7.3 National initiatives

8.1 Promoting scientific activities

8.2 Teaching - Supervision - Juries

9.1 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 Poincaré5822022HALDOI
• 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 Statistiques5432018, 1731 - 1757HALDOI
• 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 Analysis411January 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 Physics212021HALDOIback to text
• 5 articleQ.Quentin Chauleur. Growth of Sobolev norms and strong convergence for the discrete nonlinear Schrödinger equation.Nonlinear Analysis: Theory, Methods and Applications242May 2024, 113517HALDOI
• 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 Notices20244February 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 S1482021, 2877-2891HALDOI
• 9 articleC.Clément Erignoux. Hydrodynamic limit for an active exclusion process.Mémoires de la Société Mathématique de France169May 2021HALDOI
• 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 A1082August 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 2020HALDOI
• 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 Mathematics612024, 157-188In 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 Iberoamericana3712021, 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 Equations4792022, 1732-1794HALDOIback to textback to text

9.2 Publications of the year

9.3 Cited publications

• 44 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
• 45 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 Analysis411January 2021, 618--653HALDOIback to text
• 46 articleO.Oriane Blondel, C.Clément Erignoux, M.Makiko Sasada and M.Marielle Simon. Hydrodynamic limit for a facilitated exclusion process.Annales de l'IHP - Probabilités et Statistiques5612020, 667--714back to text
• 47 unpublishedG.Gilles Carbou and D.David Sanchez. Stabilization of walls in notched ferromagnetic nanowires.June 2018, PreprintHALback to text
• 48 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/rnad249DOIback to text
• 49 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.40132023, 2901--2957URL: https://doi.org/10.1007/s00220-023-04714-9DOIback to textback to text
• 50 articleS.S. De Bièvre. Relating incompatibility, noncommutativity, uncertainty, and Kirkwood--Dirac nonclassicality.Journal of Mathematical Physics642February 2023, URL: http://dx.doi.org/10.1063/5.0110267DOIback to text
• 51 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/1802.02173v12020HALDOIback to text
• 52 articleR. L.Robert L. Jerrard and D.Didier Smets. On Schrödinger maps from $T^1$ to $S^2$.Ann. Sci. ENS452012, 637-680back to text
• 53 bookD.Dan Wei. Micromagnetics and Recording Materials.http://dx.doi.org/10.1007/978-3-642-28577-6Springer--Verlag Berlin Heidelberg2012, URL: https://doi.org/10.1007/978-3-642-28577-6back to text
• 54 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