Keywords
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.2. Stochastic Modeling
 A6.1.4. Multiscale modeling
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.3. Probabilistic methods
 A6.5. Mathematical modeling for physical sciences
 B3.6. Ecology
 B3.6.1. Biodiversity
 B5.3. Nanotechnology
 B5.5. Materials
 B5.11. Quantum systems
 B6.2.4. Optic technology
1 Team members, visitors, external collaborators
Research Scientists
 Guillaume Dujardin [Team leader, INRIA, Researcher, HDR]
 Clément Erignoux [INRIA, Researcher]
 Marielle Simon [INRIA, Researcher, until Aug 2022, HDR]
Faculty Members
 Stephan De Bièvre [UNIV LILLE, Professor, HDR]
 Olivier Goubet [UNIV LILLE, Professor, HDR]
 André de Laire [UNIV LILLE, Associate Professor, HDR]
PostDoctoral Fellows
 Quentin Chauleur [UNIV LILLE, from Sep 2022]
 Lu Xu [INRIA, until Oct 2022]
 Linjie Zhao [INRIA, until Aug 2022]
PhD Students
 Christopher Langrenez [UNIV LILLE, from Sep 2022]
 Erwan Le Quiniou [UNIV LILLE, from Sep 2022]
 Anthony Nahas [UNIV LILLE, until Sep 2022]
Technical Staff
 Alexandre Roget [INRIA, Engineer]
Administrative Assistant
 Karine Lewandowski [INRIA]
2 Overall objectives
The PARADYSE team gathers mathematicians from different communities with the same motivation: to provide a better understanding of dynamical phenomena involving particles. These phenomena are described by fundamental models arising from several fields of physics. We shall focus on model derivation, study of stationary states and asymptotic behaviors, as well as links between different levels of description (from microscopic to macroscopic) and numerical methods to simulate such models. Applications include nonlinear optics, thermodynamics and ferromagnetism. Research in this direction has a long history, that we shall only partially describe in the sequel. We are confident that the fact that we come from different mathematical communities (PDE theory, mathematical physics, probability theory and numerical analysis), as well as the fact that we have strong and effective collaborations with physicists, will bring new and efficient scientific approaches to the problems we plan to tackle and will make our team strong and unique in the scientific landscape. Our goal is to obtain original and important results on a restricted yet ambitious set of problems that we develop in this document.
3 Research program
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 nonlocal problems such as the Gross–Pitaevskii (GP) equation which arises in models of dipolar gases. These models, in general, also introduce nonzero 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 nonlocal problems.
The nonlinear 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 (ULille), in the framework of the Laboratoire d’Excellence CEMPI, on its applications in nonlinear 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 40 and solitons in magnetic media are of particular interest as a mechanism for data storage or information transfer 42. It is a quasilinear PDE involving a function that takes values on the unit sphere ${\mathbb{S}}^{2}$ of ${\mathbb{R}}^{3}$. Using the stereographic projection, it can be seen as a quasilinear Schrödinger equation and the questions about the solitons, their dynamics and potential blowup 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 32, 39. In particular, the geometry of the target sphere imposes nonvanishing boundary conditions; even in dimension one, there are kinktype 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, similar models as 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 make PDE theorists and probabilists work together, in order to take advantage of the analytical results which went further ahead and are more advanced than the statistical physics theory.
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 nonlinear 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 “asymptoticpreserving” properties, energypreserving properties, or convergence to an equilibrium properties. Other numerical goals of the team include the numerical simulation of standing waves of nonlinear nonlocal 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 Application domains
4.1 Optical fibers
In the propagation of light in optical fibers, the combined effect of nonlinearity 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 nonlinear 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 nonlinearity of the fiber are periodically modulated.
As a followup 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 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 socalled Gilbert damping) and different types of anisotropic terms. The LL equation belongs to a larger class of nonlinear 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 onedimensional case and explicit formulas for solitons can be given. In the easyplane 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 groundstate 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.

Approximate models
An important physical conjecture is that the LL model is to a certain extent universal, so that the nonlinear Schrödinger and SineGordon 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.

Selfsimilar behavior
Selfsimilar solutions have attracted a lot of attention in the study of nonlinear PDEs because they can provide some important information about the dynamics of the equation. While selfsimilar expanders are related to nonuniqueness and long time description of solutions, selfsimilar shrinkers are related to a possible singularity formation. However, there is not much known about the selfsimilar 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 selfsimilar shrinkers.
This application domain involves in particular A. de Laire and G. Dujardin.
4.3 Cold atoms
The cold atoms team of the PhLAM Laboratory is reputed for having realized experimentally the socalled 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 noninteracting 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) nonlinearity. 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 nonlinear 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 nonlinear 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 and G. Dujardin.
4.4 Qualitative and quantitative properties of numerical methods
Numerical simulation of multimode fibers
The use of multimode fibers is a possible way to overcome the bandwidth crisis to come in our worldwide communication network consisting in singlemode fibers. Moreover, multimode fibers have applications in several other domains, such as high power fiber lasers and femtosecondpulse fiber lasers which are useful for clinical applications of nonlinear optical microscopy and precision materials processing. From the modeling point of view, the envelope equations are a system of nonlinear nonlocal coupled Schrödinger equations. For a better understanding of several physical phenomena in multimode fibers (e.g. continuum generation, condensation) as well as for the design of physical experiments, numerical simulations are an adapted tool. However, the huge number of equations, the coupled nonlinearities and the nonlocal effects are very difficult to handle numerically. Some attempts have been made to develop and make available efficient numerical codes for such simulations. However, there is room for improvement: one may want to go beyond MATLAB prototypes, and to develop an alternative parallelization to the existing ones, which could use the linearly implicit methods that we plan to develop and analyze. In link with the application domain 4.1, we develop in particular a code for the numerical simulation of the propagation of light in multimode fibers, using highorder efficient methods, that is to be used by the physics community.
This application domain involves in particular G. Dujardin and A. Roget.
Qualitative and longtime behavior of numerical methods
We contribute to the design and analysis of schemes with good qualitative properties. These properties may as well be “asymptoticpreserving” properties, energypreserving properties, decay properties, or convergence to an equilibrium properties. In particular, we contribute to the design and analysis of numerically hypocoercive methods for Fokker–Planck equations 37, as well as energypreserving methods for hamiltonian problems 34.
This application domain involves in particular G. Dujardin.
Highorder methods
We contribute to the design of efficient numerical methods for the simulation of nonlinear evolution problems. In particular, we focus on a class of linearly implicit highorder methods, that have been introduced for ODEs 13. We wish both to extend their analysis to PDE contexts, and to analyze their qualitative properties in such contexts.
This application domain involves in particular G. Dujardin.
4.5 Modeling of the liquidsolid transition and interface propagation
Analogously to the socalled 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 liquidsolid transition is investigated by C. Erignoux, M. Simon and their coauthors in 3 and 35. The focus of these articles is the socalled 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 steadystates of the model. For this reason, C. Erignoux, A. Roget and M. Simon are currently working with 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 C. Erignoux, A. Roget and M. Simon.
4.6 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 HautsdeFrance 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 pikeplum) which find refuge in certain forest species.
Running numerics on models coconstructed with ecologists is also at the heart of the project. The timescales of animals and plants are no different; the beetle larvae spend a few years in the earth before moving. As a byproduct, 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 advectiondiffusion 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 New software and platforms
5.1 New software
5.1.1 MM_Propagation

Name:
MultiMode Propagation

Keywords:
Optics, Numerical simulations, Computational electromagnetics

Functional Description:
This C++ software, which is interfaced with MatLab, simulates the propagation of light in multimode optical fibers. It takes into account several physical effects such as dispersion, Kerr effect, Raman effect, coupling between the modes. It uses high order numerical methods that allow for precision at reasonable computational cost.
 URL:

Contact:
Alexandre Roget
6 New results
Participants: Quentin Chauleur, Stephan De Bièvre, André de Laire, Guillaume Dujardin, Clément Erignoux, Olivier Goubet, Marielle Simon, Lu Xu, Linjie Zhao.
Some of the results presented below overlap several of the main research themes presented in section 3. However, results presented in paragraphs 6.16.6 are mainly concerned with research axis 3.1, whereas paragraphs 6.76.16 mostly concern axis 3.2. Paragraphs 6.176.19 concern numericsoriented results, and are encompassed in axis 3.3.
6.1 CrankNicolson scheme for logarithmic nonlinear Schrödinger equations with non standard dispersion
In 9, we consider a nonlinear Schrödinger equation with discontinuous modulation and logarithmic non linearity. We regularize the nonlinearity at 0 to avoid numerical problems; the regularization parameter is $\epsilon $. We analyze the consistence of the classical CrankNicolson scheme and provide precise error estimates depending on the time step $\tau $ and on ${\epsilon}^{1}$.
6.2 Standing waves for nonlinear Schrödinger equations with non standard dispersion
In 14 and in the one dimensional case we study the existence of standing waves for a nonlinear Schrödinger equation whose dispersion is singular at $x=0$. We overcome the difficulty that the problem is not invariant by space translations by introducing a suitable framework that takes into account the symetries of the problem.
6.3 Existence and decay of traveling waves for the nonlocal Gross–Pitaevskii equation
The nonlocal Gross–Pitaevskii equation is a model that appears naturally in several areas of quantum physics, for instance in the description of superfluids and in optics when dealing with thermooptic materials because the thermal nonlinearity is usually highly nonlocal. A. de Laire and S. LópezMartínez considered a nonlocal family of Gross–Pitaevskii equations in dimension one, and they found in 15 general conditions on the interactions, for which there is existence of dark solitons for almost every subsonic speed. Moreover, they established properties of the solitons such as exponential decay at infinity and analyticity. This work improves on the results obtained in P. Mennuni's PhD thesis.
6.4 Recent results for the Landau–Lifshitz equation
In 16, A. de Laire surveys recent results concerning the Landau–Lifshitz equation, a fundamental nonlinear PDE with a strong geometric content, describing the dynamics of the magnetization in ferromagnetic materials. He revisits the Cauchy problem for the anisotropic Landau–Lifshitz equation, without dissipation, for smooth solutions, and also in the energy space in dimension one. He also examines two approximations of the Landau–Lifshitz equation given by the sine–Gordon equation and the cubic Schrödinger equation, arising in certain singular limits of strong easyplane and easyaxis anisotropy, respectively. Concerning localized solutions, he reviews the orbital and asymptotic stability problems for a sum of solitons in dimension one, exploiting the variational nature of the solitons in the hydrodynamical framework. Finally, he surveys results concerning the existence, uniqueness and stability of selfsimilar solutions (expanders and shrinkers) for the isotropic LL equation with Gilbert term.
6.5 Minimizing travelling waves for the Gross–Pitaevskii equation
In 27, A. de Laire, P. Gravejat and D. Smets study the 2D Gross–Pitaevskii equation with periodic conditions in one direction, or equivalently on the product space $\mathbb{R}\times {\mathbb{T}}_{L}$ where $L>0$ and ${\mathbb{T}}_{L}=\mathbb{R}/L\mathbb{Z}.$ They focus on the variational problem consisting in minimizing the Ginzburg–Landau energy under a fixed momentum constraint. They prove that there exists a threshold value for $L$ below which minimizers are the onedimensional dark solitons, and above which no minimizer can be onedimensional.
6.6 Modulational instability in random fibers and stochastic Schrödinger equations
The team achieved an analysis of modulational instability in optical fibers with a normal dispersion perturbed with a coloured noise in 11. The effect of coloured noise on the modulational instability was investigated in order to assess whether it can produce a larger modulational instability than periodic modulations or homogeneous fibers with anomalous dispersion. They found that generally this is not the case. In 19, randomly dispersionmanaged fibers are on the contrary shown to be able to produce such large instabilities. This research was carried out with physicists from the PhLAM laboratory in Lille.
6.7 Large deviations principle for the SSEP with weak boundary interactions
Efficiently characterizing nonequilibrium stationary states (NESS) has been in recent years a central question in statistical physics. The Macroscopic Fluctuations Theory 33 developped by Bertini et al. has laid out a strong mathematical framework to understand NESS, however fully deriving and characterizing large deviations principles for NESS remains a challenging endeavour. In 20, C. Erignoux and his collaborators proved that a static large deviations principle holds for the NESS of the classical Symmetric Simple Exclusion Process (SSEP) in weak interaction with particles reservoirs. This result echoes a previous result by Derrida, Lebowitz and Speer 36, where the SSEP with strong boundary interactions was considered. In 20, it was also shown that the rate function can be characterized both by a variational formula involving the corresponding dynamical large deviations principle, and by the solution to a nonlinear differential equation. The obtained differential equation is the same as in 36, with different boundary conditions corresponding to the different scales of boundary interaction.
6.8 Mapping hydrodynamics for the facilitated exclusion and zerorange processes
In 24, we derive the hydrodynamic limit for two degenerate lattice gases, the facilitated exclusion process (FEP) and the facilitated zerorange process (FZRP), both in the symmetric and the asymmetric case. For both processes, the hydrodynamic limit in the symmetric case takes the form of a diffusive Stefan problem, whereas the asymmetric case is characterized by a hyperbolic Stefan problem. Although the FZRP is attractive, a property that we extensively use to derive its hydrodynamic limits in both cases, the FEP is not. To derive the hydrodynamic limit for the latter, we exploit that of the zerorange process, together with a classical mapping between exclusion and zerorange processes, both at the microscopic and macroscopic level. Due to the degeneracy of both processes, the asymmetric case is a new result, but our work also provides a simpler proof than the one that was previously proposed for the FEP in the symmetric case in 35.
6.9 Equilibrium perturbations for stochastic interacting systems
In 28, we consider the equilibrium perturbations for two stochastic systems: the $d$dimensional generalized exclusion process and the onedimensional chain of anharmonic oscillators. We add a perturbation of order ${N}^{\alpha}$ to the equilibrium profile, and speed up the process by ${N}^{1+\kappa}$ for parameters $0<\kappa \le \alpha $. Under some additional constraints on $\kappa $ and $\alpha $, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.
6.10 Moderate deviations for the current and tagged particle in symmetric simple exclusion processes
In 29, we prove moderate deviation principles for the tagged particle position and current in one dimensional symmetric simple exclusion processes. There is at most one particle per site. A particle jumps to one of its two neighbors at rate $1/2$, and the jump is suppressed if there is already one at the target site. We distinguish one particular particle which is called the tagged particle. We first establish a variational formula for the moderate deviation rate functions of the tagged particle positions based on moderate deviation principles from hydrodynamic limit proved by Gao and Quastel 38 Then we construct a minimizer of the variational formula and obtain explicit expressions for the moderate deviation rate functions.
6.11 The voter model with a slow membrane
In 30, we introduce the voter model on the infinite lattice with a slow membrane and investigate its hydrodynamic behavior. The model is defined as follows: a voter adopts one of its neighbors' opinion at rate one except for neighbors crossing the hyperplane $\{x:{x}_{1}=1/2\}$, where the rate is $\alpha {N}^{\beta}$. Above, $\alpha >0,\beta \ge 0$ are two parameters and $N$ is the scaling parameter. The hydrodynamic equation turns out to be heat equation with various boundary conditions depending on the value of $\beta $. The proof is based on duality method.
6.12 Longtime behavior of SSEP with slow boundary
In 17, we consider the symmetric simple exclusion process with slow boundary first introduced in 31. We prove a law of large number for the empirical measure of the process under a longer time scaling instead of the usual diffusive time scaling.
6.13 Hydrodynamics for onedimensional ASEP in contact with a class of reservoirs
In 41, we study the hydrodynamic behaviour of the asymmetric simple exclusion process (ASEP) on the lattice of size $n$, in contact with a type of slow boundary reservoirs. A scalar conservation law with boundarytrace conditions is obtained as the hydrodynamic limit in the Euler spacetime scale.
6.14 A Microscopic Derivation of Coupled SPDE’s with a KPZ Flavor
In 10, we consider an interacting particle system driven by a Hamiltonian dynamics and perturbed by a conservative stochastic noise so that the full system conserves two quantities: energy and volume. The Hamiltonian part is regulated by a scaling parameter vanishing in the limit. We study the form of the fluctuations of these quantities at equilibrium and derive coupled stochastic partial differential equations with a KPZ flavor.
6.15 Mathematical modeling for ecology
The team had an important contribution to multiscale ecosystem modeling. O. Goubet and his collaborators computed in 12 the large population limit of a stochastic process that models the evolution of a complex forest ecosystem to an evolution convectiondiffusion equation that is more suitable for concrete computations. Then, they proved on the limit equation that the existence of exchange of population between forest patches slows down the extinction of species.
6.16 Quantum optics and quantum information
Given two orthonormal bases in a ddimensional Hilbert space, one may associate to each state its Kirkwood–Dirac (KD) quasiprobability distribution. KDnonclassical states – those for which the KDdistribution takes on negative and/or nonreal values – have been shown to provide a quantum advantage in quantum metrology and information, raising the question of their identification. Under suitable conditions of incompatibility between the two bases, S. De Bièvre provided sharp lower bounds on the support uncertainty of states that guarantee their KDnonclassicality in 4 and 22. In particular, when the bases are completely incompatible, a new notion introduced in this work, states whose support uncertainty is not equal to its minimal value d+1 are necessarily KDnonclassical. The implications of these general results for various commonly used bases, including the mutually unbiased ones, and their perturbations, are detailed.
In quantum optics, the notion of classical state is different than in the discrete value systems described above. In that case one requires the GlauberSudarshan Pfunction to be positive. Characterizing the classical states is a longstanding problem in this context as well. In 25, S. De Bièvre and collaborators establish an interferometric protocol allowing to determine a recently introduced nonclassicality measure, known as the quadrature coherence scale (QCS) 7. A detailed study of the QCS of photonadded/subtracted states is provided in 26.
6.17 Linearly implicit highorder numerical methods for evolution problems
G. Dujardin and his collaborator derived in 13 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. 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 are expected 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.
6.18 Numerical simulation of multispecies Bose–Einstein condensates
In 23, G. Dujardin, A. Nahas and I. LacroixViolet proposed a new numerical method for the simulation of multicomponent Bose–Einstein condensates in dimension 2. They implemented their method and demonstrated its efficiency compared to existing methods from the literature, in several physically relevant regimes (vortex nucleation, vortex sheets, giant holes, etc were obtained numerically). They verified numerically several theoretical results known for the minimizers in strong confinment regimes. They also supported numerically theoretical conjectures in other physically relevant contexts. In addition, they developped postprocessing algorithms for the automatic detection of vortex structures (simple vortices, vortex sheets, etc), as well as for the numerical computation of indices.
6.19 Discrete quantum harmonic oscillator and Kravchuk transform
We consider in 21 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 secondorder 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.
7 Partnerships and cooperations
Participants: Stephan De Bièvre, André de Laire, Guillaume Dujardin, Clément Erignoux, Olivier Goubet, Marielle Simon.
7.1 International initiatives
Projet LISA IEA CNRS LilleSantiago
Participants: André de Laire, Olivier Goubet.
 Title: LISA

Partner Institution(s):
Laboratoire Paul Painlevé, Lille, CNRS, and CMM, UMI CNRS, Santiago du Chili

Date/Duration:
20202022

Nature of the initiative:
International Emerging Action (IEA) CNRS

Budget:
4000 €/year

Objective:
Study of dispersive equations
7.2 International research visitors
7.2.1 Visits of international scientists
Claudio Muñoz

Status
Researcher

Institution of origin:
University of Chile

Country:
Chile

Dates:
Dec. 89, 2022

Context of the visit:
projet LISA IEA CNRS LilleSantiago

Mobility program/type of mobility:
research stay
7.2.2 Visits to international teams
André de Laire

Visited institution:
University of Birmingham

Country:
United Kingdom

Dates:
Oct. 612, 2022

Context of the visit:
Research projet with Susana Gutiérrez

Mobility program/type of mobility:
research stay
Guillaume Dujardin

Visited institution:
University of Gothenburg (Chalmers)

Country:
Sweden

Dates:
Dec. 59, 2022

Context of the visit:
Research projet with David Cohen and Andre Berg

Mobility program/type of mobility:
research stay
Clément Erignoux

Visited institution:
Instituto Superior Técnico

Country:
Portugal

Dates:
May 26, 2022

Context of the visit:
Research projet with Patricia Gonçalves and Gabriel Nahum

Mobility program/type of mobility:
research stay
Clément Erignoux

Visited institution:
Cambridge University

Country:
UK

Dates:
May 1618, 2022

Context of the visit:
Research projet with Maria Bruna, Robert Jack and James Mason

Mobility program/type of mobility:
research stay
7.3 European initiatives
 Clément Erignoux and Marielle Simon are part of a francoportuguese Pessoa project
 Title: "Kinetic theory, particle systems and their hydrodynamic limits".
 Grant 2022: 2400€
7.4 National initiatives
7.4.1 ANR MICMOV
 Marielle Simon is the PI of the ANR MICMOV project
 Title: “Microscopic description of moving interfaces”
 Link to the website
 ANR Reference: ANR19CE400012
 Members: M. Simon (PI, Inria Lille), G. Barraquand (LPTENS Paris), O. Blondel (Université de Lyon), C. Cancès (Inria Lille), C. Erignoux (Inria Lille), M. Herda (Inria Lille), L. Zhao (Inria Lille)
 Total amount of the grant: 132 000 euros
 Duration: March 2020 – October 2024
7.4.2 LabEx CEMPI
Through their affiliation to the Laboratoire Paul Painlevé of Université de Lille, PARADYSE team members benefit from the support of the Labex CEMPI. In addition, the LabEx CEMPI is funding the postdoc of Quentin Chauleur in the team, in an interdisciplinary initiative between PhLAM and LPP.
 Title: Centre Européen pour les Mathématiques, la Physique et leurs Interactions
 Partners: Laboratoire Paul Painlevé (LPP) and Laser Physics department (PhLAM), Université de Lille
 ANR reference: 11LABX0007
 Duration: February 2012  December 2024 (the project has been renewed in 2019)
 Budget: 6 960 395 euros
 Coordinator: Emmanuel Fricain (LPP, Université de Lille)
The "Laboratoire d'Excellence" CEMPI (Centre Européen pour les Mathématiques, la Physique et leurs Interactions), a project of the Laboratoire de mathématiques Paul Painlevé (LPP) and the laboratoire de Physique des Lasers, Atomes et Molécules (PhLAM), was created in the context of the "Programme d'Investissements d’Avenir" in February 2012. The association PainlevéPhLAM creates in Lille a research unit for fundamental and applied research and for training and technological development that covers a wide spectrum of knowledge stretching from pure and applied mathematics to experimental and applied physics. The CEMPI research is at the interface between mathematics and physics. It is concerned with key problems coming from the study of complex behaviors in cold atoms physics and nonlinear optics, in particular fiber optics. It deals with fields of mathematics such as algebraic geometry, modular forms, operator algebras, harmonic analysis, and quantum groups, that have promising interactions with several branches of theoretical physics.
7.5 Regional initiatives
The PARADYSE projectteam was granted the SIMPAPH “Action de Développement Technologique”, which allowed to hire Alexandre Roget as an engineer in the projectteam from 2019 to 2021. We obtained extensions of the contract of A. Roget, which was a teammember in 2022 as well. This ADT SIMPAPH's goals were originally threefold:

 develop a software for the simulation of the propagation of light in multimode optical fibers for the optical physics community;
 simulate large systems of random particles such as twodimensional constrained lattice gases;
 simulate the dynamics of 3D Bose–Einstein condensates.
8 Dissemination
Participants: Stephan De Bièvre, André de Laire, Guillaume Dujardin, Clément Erignoux, Olivier Goubet, Marielle Simon.
8.1 Promoting scientific activities
8.1.1 Scientific events: organisation
 Workshop "Asymetry in interacting particle systems", held at INRIA Lille on October 35 2022.
 Organizers: G. Barraquand, O. Blondel, C. Erignoux, P. Illien, M. Simon
 "Journée de rentrée de l'équipe probasstats", held at Lille university on September 22 2022.
 Organizers: C. Baey, S. Dabo, D. Dereudre, C. Duval, C. Erignoux, B. Thiam
 "Journée Analyse Appliquée Hautsde France" 10 novembre 2022.
 Organizers: C. Calgaro, O. Goubet, M. Herda
 "Journée des Doctorants en Mathématiques de la région HautsdeFrance", held at Université Polytechnique HautsdeFrance on September 9, 2022.
 Organizers: S. Biard, M. Davila, A. de Laire, A. El Mazouni, R. Ernst, B. Testud
8.1.2 Journal
Member of the editorial boards:
$\phantom{\rule{0.277778em}{0ex}}$ S. De Bièvre is associate editor of the Journal of Mathematical Physics (since January 2019). $\phantom{\rule{0.277778em}{0ex}}$ O. Goubet is the editor in chief of the NorthWestern European Journal of Mathematics. $\phantom{\rule{0.277778em}{0ex}}$ O. Goubet is associate editor of ANONA (Advances in Nonlinear Analysis) $\phantom{\rule{0.277778em}{0ex}}$ O. Goubet is associate editor of the Journal of Math. Study.
Reviewer  reviewing activities:
$\phantom{\rule{0.277778em}{0ex}}$ All permanent members of the PARADYSE team work as referees for many of the main scientific publications in analysis, probability and statistical physics, depending on their respective fields of expertise.
8.1.3 Invited talks
All PARADYSE team members take active part in numerous scientific conferences, workshops and seminars, and in particular give frequent talks both in France and abroad.
8.1.4 Research administration
 S. De Bièvre and A. de Laire are both members of the “Conseil de Laboratoire Paul Painlevé” at Université de Lille.
 S. De Bièvre is member of the executive committee of the LabEx CEMPI.
 A. de Laire is member of the “Fédération de Recherche Mathématique des HautsdeFrance”.
 G. Dujardin is a member of the Executive Committee of the CPER Wavetech.
 O. Goubet is member of the "conseil de département de mathématiques" at Université de Lille.
 O. Goubet is member of "bureau du HUB numérique" de ISite ULille
 O. Goubet is the president of SMAI (Société de Mathématiques Appliquées et Industrielles)
 C. Erignoux is a member of the LNE Inria research center's "Comité de Centre".
8.2 Teaching  Supervision  Juries
8.2.1 Teaching
The PARADYSE team teaches various undergraduate level courses in several partner universities and Grandes Écoles. We only make explicit mention here of the Master courses (level M1M2) and the doctoral courses.
 Master: O. Goubet and A. de Laire, "Modélisation et Approximation par Différences Finies", M1 (Université de Lille, 54h).
 Master: O. Goubet, "Etude de problèmes elliptiques et paraboliques", M1 (Université de Lille, 24h).
 Master: A. de Laire, "Analyse numérique pour les EDP", M1 (Université de Lille, 60h).
 Master: C. Erignoux and L. Xu, "Advanced probabilites", M2 (Université de Lille, 24h).
 Master : G. Dujardin, "Condensats de Bose–Einstein : théorie et simulation numérique" (Université de Lille, 24h).
 Doctoral School: M. Simon, "Harmonic chain of oscillators with random flips of velocities" (GSSI Institute, L'Aquila, Italy, 12h).
 Doctoral School: S. De Bièvre, "Quantum information" (Université de Lille, 24h).
S. De Bièvre has represented (20182021) the department of Mathematics in the organization of the newly created Master of Data Science of EC Lille, Université de Lille and IMT. This role has since been taken over by O. Goubet.
8.2.2 Supervision
 C. Erignoux and M. Simon supervised Adel Assakaf, M2  Maths internship, "Effet de dynamiques de bord sur le FEP". (10 weeks)
 C. Erignoux supervised Hugo Dorfsman, M1  Calcul scientifique internship, "Comportement microscopique et macroscopique du SSEP en interaction faible avec des reservoirs". (10 weeks)
 C. Erignoux supervised Fael Rebei, L3  ENS internship, "Limite hydrodynamique et grandes deviations pour le SSEP". (6 weeks)
 S. De Bièvre supervised C. Langrenez, M2maths internship, “KD nonclassicality.” (12 weeks)
 S. De Bièvre is supervising V. Niaussat, M2maths internship, “Quantum random sampling.” (12 weeks)
 S. De Bièvre is supervising the PhD thesis of C. Langrenez on “KD nonclassicality”. 20222024.
 G. Dujardin supervised Charbel Ghosn, M2maths internship, "High order numerical methods for high dimensional systems of ODEs". (12 weeks)
 G. Dujardin supervised (with I. LacroixViolet) the PhD thesis 18 of Anthony Nahas, entitled "Simulation numérique de condensats de Bose–Einstein", which was successfully defended in October 2022.
 A. de Laire and O. Goubet supervised Erwan Le Quiniou, M2  Maths internship, "Solitons for a quasilinear GrossPitaevskii equation". (12 weeks)
 A. de Laire and O. Goubet are supervising the PhD thesis of Erwan Le Quiniou, "Solitons for the Landau–Lifshitz equation". 20222025.
8.2.3 Juries
 G. Dujardin served as reviewer for the PhD thesis of Martino Lovisetto (University of Nice, May 10th, 2022), entitled "Theoretical and numerical study of the SchrödingerNewton equation with application to nonlinear optics and experimental observation of violent relaxation", supervised by Dider Clamond and Bruno Marcos.
 G. Dujardin served as reviewer for the PhD thesis of Grégoire Barrué (ÉNS de Rennes, July 7th, 2022), entitled "Approximation diffusion pour des équations dispersives", supervised by Arnaud Debussche and Anne De Bouard.
8.3 Popularization
8.3.1 Articles and contents
A. de Laire made a contribution to the "Book Summaries" section of the magazine Matapli N°127 by SMAI (Society of Applied and Industrial Mathematics).
8.3.2 Interventions
A. de Laire participed to the meeting Declics 2022 (Dialogue Between Researchers and High School Students to Engage them in the Construction of Knowledge) at Lycée Faidherbe, Lille
9 Scientific production
9.1 Major publications
 1 articleA Microscopic Derivation of Coupled SPDE's with a KPZ Flavor.Annales de l'Institut Henri Poincaré5822022
 2 articleEnergy preserving methods for nonlinear Schrödinger equations.IMA Journal of Numerical Analysis411January 2021, 618–653
 3 articleStefan problem for a nonergodic facilitated exclusion process.Probability and Mathematical Physics212021
 4 articleComplete Incompatibility, Support Uncertainty, and KirkwoodDirac Nonclassicality.Physical Review Letters2021
 5 articleDecay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion.Discrete and Continuous Dynamical Systems  Series S1482021, 28772891
 6 articleHydrodynamic limit for an active exclusion process.Mémoires de la Société Mathématique de France169May 2021
 7 articleQuadrature coherence scale driven fast decoherence of bosonic quantum field states.Physical Review LettersMarch 2020
 8 articleExistence and decay of traveling waves for the nonlocal GrossPitaevskii equation.Communications in Partial Differential Equations4792022, 17321794
9.2 Publications of the year
International journals
 9 articleCrankNicolson scheme for a logarithmic Schrödinger equation.NorthWestern European Journal of Mathematics2022
 10 articleA Microscopic Derivation of Coupled SPDE's with a KPZ Flavor.Annales de l'Institut Henri Poincaré5822022
 11 articleStochastic modulational instability in the nonlinear Schrödinger equation with colored random dispersion.Physical Review AJanuary 2022
 12 articleComparison principles and applications to mathematical modelling of vegetal metacommunities.Mathematics in Engineering452022, 117
 13 articleHigh order linearly implicit methods for evolution equations: How to solve an ODE by inverting only linear systems.ESAIM: Mathematical Modelling and Numerical AnalysisApril 2022
 14 articleStanding waves for semilinear Schrödinger equations with discontinuous dispersion.Rendiconti del Circolo Matematico di Palermo713December 2022, 11591171
 15 articleExistence and decay of traveling waves for the nonlocal GrossPitaevskii equation.Communications in Partial Differential Equations4792022, 17321794
 16 articleRecent results for the LandauLifshitz equation.SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada7922022, 253295
 17 articleLongtime behavior of SSEP with slow boundary.Statistics and Probability Letters2022
Doctoral dissertations and habilitation theses
 18 thesisNumerical simulation of BoseEinstein condensates.Université de LilleOctober 2022
Reports & preprints
 19 miscModulational instability in randomly dispersionmanaged fiber links.December 2022
 20 miscSteady state large deviations for onedimensional, symmetric exclusion processes in weak contact with reservoirs.December 2022
 21 miscDiscrete quantum harmonic oscillator and kravchuk transform.December 2022
 22 miscRelating incompatibility, noncommutativity, uncertainty and KirkwoodDirac nonclassicality.December 2022
 23 miscA numerical study of vortex nucleation in 2D rotating BoseEinstein condensates.November 2022
 24 miscMapping hydrodynamics for the facilitated exclusion and zerorange processes.December 2022
 25 miscInterferometric measurement of the quadrature coherence scale using two replicas of a quantum optical state.December 2022
 26 miscNonclassicality gain/loss through photonaddition/subtraction on multimode Gaussian states.December 2022

27
miscMinimizing travelling waves for the GrossPitaevskii equation on
$$ .January 2022  28 miscEquilibrium perturbations for stochastic interacting systems.June 2022
 29 miscModerate deviations for the current and tagged particle in symmetric simple exclusion processes.March 2022
 30 miscThe voter model with a slow membrane.March 2022
9.3 Cited publications
 31 articleExclusion process with slow boundary.Journal of Statistical Physics1672017, 1112–1142

32
articleGlobal Schrödinger maps in dimensions
$d2$ : small data in the critical Sobolev spaces.Annals of Mathematics2011, 14431506  33 articleMacroscopic fluctuation theory.Reviews of Modern Physics872Jun 2015, 593–636URL: http://dx.doi.org/10.1103/RevModPhys.87.593
 34 articleEnergy preserving methods for nonlinear Schrödinger equations.IMA Journal of Numerical Analysis411January 2021, 618653
 35 articleHydrodynamic limit for a facilitated exclusion process.Annales de l'IHP  Probabilités et Statistiques5612020, 667714
 36 articleLarge Deviation of the Density Profile in the Steady State of the Open Symmetric Simple Exclusion Process.Journal of Statistical Physics1073/42002, 599–634URL: http://dx.doi.org/10.1023/A:1014555927320
 37 articleCoercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker Planck equations.Numerische Mathematik144https://arxiv.org/abs/1802.02173v12020
 38 articleModerate deviations from the hydrodynamic limit of the symmetric exclusion process.Science in China Series A: Mathematics4652003, 577–592

39
articleOn Schrödinger maps from
${T}^{1}$ to${S}^{2}$ .Ann. Sci. ENS452012, 637680  40 bookMicromagnetics and Recording Materials.http://dx.doi.org/10.1007/9783642285776SpringerVerlag Berlin Heidelberg2012, URL: https://doi.org/10.1007/9783642285776
 41 articleHydrodynamics for OneDimensional ASEP in Contact with a Class of Reservoirs.Journal of Statistical Physics1891October 2022, 1
 42 articleTopological computation based on direct magnetic logic communication.Scientific Reports52015, URL: http://dx.doi.org/10.1038/srep15773