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, Starting Faculty Position]
 Marielle Simon [Inria, Researcher, HDR]
Faculty Members
 Stephan De Bièvre [Université de Lille, Professor, HDR]
 Olivier Goubet [Université de Lille, Professor, HDR]
 André de Laire [Université de Lille, Associate Professor, HDR]
PostDoctoral Fellows
 Salvador López Martínez [Inria, until Aug 2021]
 Linjie Zhao [Inria]
PhD Student
 Anthony Nahas [Université de Lille]
Technical Staff
 Alexandre Roget [Inria, Engineer]
Interns and Apprentices
 Hugues Moyart [Ecole normale supérieure ParisSaclay, from Apr 2021 until Jul 2021]
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 39 and solitons in magnetic media are of particular interest as a mechanism for data storage or information transfer 41. 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 28, 35. 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 atoms (precisely chains of oscillators) with different local microscopic defects. We apply our recent techniques to understand how anomalous (in particular fractional) diffusive systems interact with the boundaries. For instance, the powerful tool given by Wigner functions that we already used has been successfully applied to the derivation of anomalous behaviors in open systems (for instance in 34). The next step consists in developing an extension of that tool to deal with bounded systems provided with fixed boundaries. We also intend to derive anomalous diffusion by adding longrange interactions to diffusive models. There are very few rigorous results in this direction.
Finally, we aim at obtaining from a microscopic description the fractional porous medium equation (FPM), a nonlinear variation of the fractional diffusion equation, involving the fractional Laplacian instead of the usual one. Its rigorous study carries many mathematical difficulties in treating at the same time the nonlinearity and fractional diffusion. We want 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 33, as well as energypreserving methods for hamiltonian problems 2.
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 23. 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 31. 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 trying to work 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 results
Participants: Stephan De Bièvre, André de Laire, Guillaume Dujardin, Clément Erignoux, Olivier Goubet, Salvador LópezMartínez, Marielle Simon, Linjie Zhao.
Some of the results presented below overlap several of the main research themes presented in section 3. However, results presented in paragraphs 5.15.5 are mainly concerned with research axis 3.1, whereas paragraphs 5.65.13 mostly concern axis 3.2. Paragraphs 5.145.16 concern numericsoriented results, and are encompassed in axis 3.3.
5.1 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 26 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 by A. de Laire and P. Mennuni in 38.
5.2 The cubic Schrödinger regime of the Landau–Lifshitz equation with a strong easyaxis anisotropy
It is wellknown that the dynamics of biaxial ferromagnets with a strong easyaxis anisotropy is essentially governed by the cubic Schrödinger equation. A. de Laire and P. Gravejat provided in 7 a rigorous justification to this observation, continuing with the work started in 37. More precisely, they showed the convergence of the solutions to the Landau–Lifshitz equation for biaxial ferromagnets towards the solutions to the cubic Schrödinger equation in the regime of an easyaxis anisotropy. This result holds for solutions to the Landau–Lifshitz equation in highorder Sobolev spaces. By introducing highorder energy quantities with good symmetrization properties, they derived the convergence from the consistency of the Landau–Lifshitz equation with the sineGordon equation by using welltailored energy estimates.
In this regime, they additionally classified the onedimensional solitons of the Landau–Lifshitz equation and quantified their convergence towards the solitons of the onedimensional cubic Schrödinger equation.
5.3 Recent results for the Landau–Lifshitz equation
In 17, 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 sineGordon 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.
5.4 Modulational instability in random fibers and stochastic Schrödinger equations
The team achieved an analysis of modulational instability in optical fibers with randomly kicked normal dispersion in 12 as well as with a normal dispersion perturbed with a coloured noise in 20. S. De Bièvre, G. Dujardin and collaborators developed and analyzed in 12 a physically realistic model of optical fibers with randomly kicked normal dispersion. They analyzed the modulational instability generated in such fibers through the associated gain, both theoretically and numerically. In 20 the effect of coloured noise on the modulational instability was investigated in order to assess whether it can produce a larger modulational instability. We found that generally this is not the case. This research was carried out with physicists from the PhLAM laboratory in Lille.
Another important result of the team in this direction is 5, where O. Goubet et al. established the decay rate of solutions to a nonlinear Schrödinger equation with stochastic modulation. The result of 5 is the first result concerning the long time behavior of solutions for nonlinear Schrödinger equations with white noise modulation. The final result is that the decay rate towards equilibrium for the nonlinear Schödinger equation is twice slower with white noise modulation than in the deterministic case.
5.5 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. 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.
5.6 Hydrodynamic limit for a chain with thermal and mechanical boundary forces
In a collaboration with T. Komorowski and S. Olla 15, M. Simon proved the hydrodynamic limit for an harmonic chain with a random exchange of momentum that conserves the kinetic energy but not the momentum. The system is open and subject to two thermostats at the boundaries and to external tension. Under a diffusive scaling of spacetime, the authors proved that the empirical profiles of the two locally conserved quantities, the volume stretch and the energy, converge to the solution of a nonlinear diffusive system of conservative partial differential equations.
5.7 Stefan problem for a nonergodic facilitated exclusion process
In 3, O. Blondel, C. Erignoux and M. Simon investigated the general hydrodynamics for the facilitated exclusion process whose supercritical phase's hydrodynamics had been previously investigated in 31. This process is similar to the celebrated symmetric simple exclusion process, except that a particle is only allowed to jump to a neighboring site if its other neighbor is occupied by a particle. This hard constraint on the particle's motion has a number of consequences on the microscopic and macroscopic behavior of the system. In particular, under the critical density ${\rho}_{c}=1/2$, the system quickly freezes out and particles stop moving.
The purpose of this work was to investigate the macroscopic invasion of the frozen phase by the ergodic phase, and the authors were able to prove that starting from a profile with both supercritical and subcritical regions, the hydrodynamics for the facilitated exclusion process is given by a Stefan problem: the diffusive supercritical phase progressively invades the subcritical phase via flat interfaces, until either one of the phases disappears.
5.8 Hydrodynamic limit for an active exclusion process
In 6, C. Erignoux investigated the scaling limit of an active exclusion process, which is a microscopic dynamics put forward to model selforganization as observed in various animal species (e.g. school of fish, flock of birds). Active models can exhibit rich phenomenology, such as Motility Induced Phase Separation (MIPS), and the formation of spontaneous collective dynamics, as in Viszek's celebrated model. However, because of the exclusion rule between particles (two particles cannot occupy the same site of the lattice), the model is nongradient, resulting in a complex, nonexplicit, crossdiffusive hydrodynamic equation.
In 24, C. Erignoux builds on previous work 6, 36 and explores various aspects of modeling individualbased active matter models. In particular, 24 describes general aspects of the mathematical theory of hydrodynamic limits, and basic principles to conjecture the hydrodynamic equation given the underlying microscopic system. One of the objectives of the article is to provide members of the physics community interested in active matter with tools to navigate and use mathematical theory for hydrodynamic limits. The article also conjectures the socalled nonequilibrium fluctuating hydrodynamics for active lattice gases. The long term goal is to derive in a mathematically satisfying way the two phenomena described above (MIPS and collective dynamics), for which the hydrodynamic equation yields some information, but the fluctuating hydrodynamics is necessary to fully understand the underlying mechanisms.
5.9 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 30 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 27, 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 32, where the SSEP with strong boundary interactions was considered. In 27, 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 32, with different boundary conditions corresponding to the different scales of boundary interaction.
5.10 Asymmetric attractive zerorange process with destruction at the origin
In 25, C. Erignoux, M. Simon and L. Zhao considered the effect of boundary interactions on the onedimensional asymmetric zerorange process on the full line. More precisely, particles are destroyed at the origin of the system at rate $\alpha {N}^{\beta}$, and they showed that depending on the value of $\beta $, different behaviors can be derived for the macroscopic limit of the system: for negative $\beta $, particle destructions do not have a macroscopic effect, and the system macroscopically behaves as the asymmetric zerorange process without destruction. For $\beta =0$, a proportion (depending on $\alpha $) of particles is destroyed, and the righthand side $x>0$ behaves as a zerorange process with a boundary condition at the origin which is a function of the density on the left of the origin. For $\beta >0$, finally, most particles are destroyed while they go through the origin, so that no mass crosses through the origin at the macroscopic level.
5.11 SSEP with a slow bond and site boundary
In 18, L. Zhao and his coauthor considered the onedimensional symmetric simple exclusion process with a slow bond. In this model, particles cross each bond at rate ${N}^{2}$, except one particular bond, the slow bond, where the rate is $N$. Above, $N$ is the scaling parameter. This model has been considered in the context of hydrodynamic limits, fluctuations and large deviations. They investigated moderate deviations from hydrodynamics and obtained a moderate deviations principle.
5.12 Nonequilibrium fluctuations of the weakly asymmetric normalized binary contact path process
In 19, X. Xue and L. Zhao further investigated the problem studied in 40, where the authors proved a law of large numbers for the empirical measure of the weakly asymmetric normalized binary contact path process on ${\mathbb{Z}}^{d}$, $d\ge 3$, and then conjectured that a central limit theorem should hold under a nonequilibrium initial condition. They proved that the said conjecture is true when the dimension $d$ of the underlying lattice and the infection rate $\lambda $ of the process are sufficiently large.
5.13 Mathematical modeling for ecology
The team had an important contribution to the multiscale ecosystem modeling. O. Goubet and his collaborators computed in 22 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.
In 21 O. Goubet and his collaboraters addressed the initial value problem for a shallow water system of equations with a Coriolis force term. This result is nonstandard due to the Coriolis term.
5.14 Numerical integration of the stochastic Manakov system
The stochastic Manakov system is a dispersive nonlinear system of PDEs that models the propagation of light in an optical fiber with randomly varying birefringence.
In 29, G. Dujardin and his collaborators introduced a linearly implicit scheme for the time integration of the stochastic Manakov system, that they analyzed and compared to the existing methods from the literature. In particular, they proved that the strong order of the numerical approximation is $1/2$ if the nonlinear term in the system is globally Lipschitzcontinuous. They also proved that this numerical method converges with order $1/2$ in probability and with order $1/{2}^{}$ almost surely, in the case of the cubic nonlinear coupling which is relevant in optical fibers. They also proposed a modification of their method to obtain a masspreserving scheme.
In 8, G. Dujardin and his collaborators developed, analyzed and implemented a numerical method based on the Lie–Trotter formula for the integration of the stochastic Manakov system. In particular, they proved that the strong order of the numerical approximation is $1/2$ if the nonlinear term in the system is globally Lipschitz. They also proved that this splitting scheme converges with order $1/2$ in probability, and converges almost surely with order $1/{2}^{}$ as well. They provided numerical experiments to compare the efficiency of this scheme with existing methods from the literature, and they investigated numerically the possible blowup in finite time of solutions to this SPDE system.
5.15 Linearly implicit highorder numerical methods for evolution problems
G. Dujardin and his collaborator derived in 23 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.
5.16 Energypreserving methods for nonlinear Schrödinger equations
G. Dujardin and his coauthors revisited and extended relaxation methods for nonlinear Schrödinger equations (NLS). The classical relaxation method for NLS is a mass and energypreserving method. Moreover, it is only linearly implicit. A first proof of the secondorder accuracy was achieved in 2. Moreover, the method was extended to enable to treat noncubic nonlinearities, nonlocal nonlinearities, as well as rotation terms. The resulting methods are still masspreserving and energypreserving. Moreover, they are shown to have secondorder accuracy numerically. These new methods are compared with fully implicit, mass and energypreserving methods of Crank and Nicolson.
6 Partnerships and cooperations
Participants: André de Laire, Olivier Goubet, Marielle Simon.
6.1 International initiatives
 André de Laire and Olivier Goubet got a support from CNRS to develop
a collaboration on the study of travelling wave solutions with the
CMM of Santiago in Chile
(with the team of Claudio Muñoz).
 Title: "LISA (LIlleSantiago)"
 Members: A. de Laire, O. Goubet
 Total amount of the grant: 4 000 euros/year
 Duration: 20212022
6.2 European initiatives
 Marielle Simon is the PI of the MATMOVIN project cofunded by the European Union together with the “Fonds de Développement Régional”.
 Title: “Description microscopique des transitions de phase et interfaces mobiles : avancées mathématiques”
 Type: Postdoc grant of 2 years
 Duration: September 2020 – August 2022
 Research group: M. Simon (PI, Inria Lille), A. Roget (Inria Lille), L. Zhao (Inria Lille)
6.3 National initiatives
6.3.1 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
6.3.2 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.
 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.
6.3.3 ADT SIMPAPH
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. 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.
6.4 Regional initiatives
 Olivier Goubet (PI) got a support from Région HautsdeFrance (grant STIMULE STIR) to initiate a research program involving applied mathematicians in Amiens, Calais, Lille and Valenciennes.
 Title: "Super QUantum fluids and shAllow Water equations"
 Members: C. Calgaro, O. Goubet, T. Rey (ULille), J.P. Chehab, V. Desveaux, Y. Mammeri, V. Martin, H. Le Meur (UPJV), A. Benoit, C. Bourel, C. Rosier, L. Rosier (ULCO), E. Creusé (UPHF)
 Total amount of the grant: 14 400 euros
 Duration: November 2021–April 2023
7 Dissemination
Participants: Stephan De Bièvre, André de Laire, Guillaume Dujardin, Clément Erignoux, Olivier Goubet, Salvador LópezMartínez, Marielle Simon, Linjie Zhao.
7.1 Promoting scientific activities
7.1.1 Scientific events: organisation
 A. de Laire coorganized the “Journée des Doctorants en Mathématiques de la région HautsdeFrance”, on October 1st, 2021, held at the Universtié Picardie Jules Verne, Amiens. Event’s webpage.
 C. Erignoux coorganized the “Journée de rentrée du Laboratoire Paul Painlevé”, on November 25th, 2021, held at "La Piscine de Roubaix". Event’s webpage.
 O. Goubet coorganized the "Conférence en l'honneur de Serge Nicaise", from November 2nd to 5th, 2021, held at the University of Valenciennes. Event’s webpage.
 M. Simon coorganized the "Online junior conference on random graphs and interacting particle systems", from September 6th to 10th, 2021, held online. Event’s webpage.
7.1.2 Journal
Member of the editorial boards:
O. Goubet was the guest coeditor of the special issue of Discrete $\&$ Continuous Dynamical Systems  A (DCDSA) Vol. 14, no. 8 of August 2021.
S. De Bièvre is associate editor of the Journal of Mathematical Physics (since January 2019).
Reviewer  reviewing activities:
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.
7.1.3 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.
7.1.4 Leadership within the scientific community
O. Goubet is the president of the Société de Mathématiques Appliquées et Industrielles (SMAI).
7.1.5 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.
 C. Erignoux is a member of the LNE Inria research center's "Comité de Centre".
 M. Simon is member of the CNU (Conseil National des Universités), Section 26.
7.2 Teaching  Supervision  Juries
7.2.1 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).
 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 represents (since 2018) the department of Mathematics in the organization of the newly created Master of Data Science of EC Lille, Université de Lille and IMT.
7.2.2 Supervision
 A. de Laire supervised the postdoc of S. LópezMartínez, which ended in August 2021.
 G. Dujardin coadvises (with I. LacroixViolet) the PhD thesis of Anthony Nahas. Title: "Simulation of rotating multispecies Bose–Einstein condensates".
 C. Erignoux and M. Simon cosupervise the postdoc of Linjie Zhao.
 C. Erignoux supervised the first year of master's internship of Hugues Moyart from March to August 2021. Title: "Duality and influence of thermostats on the macroscopic limit of the symmetric simple exclusion process".
 O. Goubet coadvises (with B. Alouini, B. Dehman and V. Martin) the PhD thesis of Mariem Abidi. Title: "Logarithmic Schrödinger equations".
 O. Goubet coadvises (with V. Desveaux) the PhD thesis of Alice Masset. Title: "Shallow water equations with Coriolis forcing and temperature".
 O. Goubet coadvises (with G. Decocq) the PhD thesis of Clément Carlier. Title: "Models for metapopulations in forest ecology".
 M. Simon coadvises (with P. Gonçalves) the PhD thesis of Gabriel Nahum. Title: "Nonlinear Problems in Interacting Particle Systems".
7.2.3 Juries
 A. de Laire was referee and member of the jury of the PhD thesis of J. Alhelou (Université de Toulouse III, November 2021). Title: "Mathematical and numerical analysis for a Gross–Clark–Schrödinger system".
 A. de Laire was referee and member of the jury of the PhD thesis of X. Yuan (École polytechnique, Paris, June 2021). Title: "Long time dynamics for nonlinear wavetype equations with or without damping".
 O. Goubet was the referee and member of the jury of the PhD thesis of S. Bahrouni (University of Monastir, Tunisia, July 2021). Title: "Orlicz–Sobolev fractional spaces and applications to nonlinear problems".
 O. Goubet was president of the jury of the PhD thesis of M. Handa (Unheld iverstié Picardie Jules Verne, Amiens, July 2021). Title: "Modelisation, optimisation and simulation of power distribution networks".
 O. Goubet was member of the jury of the Habilitation of B. Alouini (University of Monastir, Tunisia, July 2021). Title: "Study of nonlinear anisotropic Bose–Einstein and Shrödinger equations".
 M. Simon was member of the jury of the PhD thesis of B. Dagallier (École polytechnique, Paris, September 2021). Title: "Large deviations in interacting particle systems: out of equilibrium correlations and interface dynamics".
 M. Simon was member of the jury of the PhD thesis of A. Ertul (Université Lyon 1, December 2021). Title: "Diffusion and relaxation for particle systems with kinetic constraints".
 M. Simon was member of the jury of the PhD thesis of A. Hannani (Université PSL Dauphine, Paris, December 2021). Title: "Random perturbation of certain interacting particle systems related to quantum mechanics".
7.3 Popularization
A. de Laire participated in Declics 2021, a scientific speed meeting with high school students at Lycée Faidherbe, on December 7th 2021, Lille.
S. De Bièvre published a series of three articles in Images des Mathématiques on quantum cryptography.
8 Scientific production
8.1 Major publications
 1 articleInterpolation process between standard diffusion and fractional diffusion.Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques5432018, 1731  1757
 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 articleThe cubic Schrödinger regime of the LandauLifshitz equation with a strong easyaxis anisotropy.Revista Matemática Iberoamericana3712021, 95128
8.2 Publications of the year
International journals
 8 articleLieTrotter Splitting for the Nonlinear Stochastic Manakov System.Journal of Scientific Computing886May 2021
 9 articleEnergy preserving methods for nonlinear Schrödinger equations.IMA Journal of Numerical Analysis411January 2021, 618–653
 10 articleStefan problem for a nonergodic facilitated exclusion process.Probability and Mathematical Physics212021
 11 articleComplete Incompatibility, Support Uncertainty, and KirkwoodDirac Nonclassicality.Physical Review Letters2021
 12 articleModulational instability in optical fibers with randomlykicked normal dispersion.Physical Review A1035May 2021, 053521
 13 articleDecay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion.Discrete and Continuous Dynamical Systems  Series S1482021, 28772891
 14 articleHydrodynamic limit for an active exclusion process.Mémoires de la Société Mathématique de France169May 2021
 15 articleHydrodynamic limit for a chain with thermal and mechanical boundary forces.Electronic Journal of Probability262021
 16 articleThe cubic Schrödinger regime of the LandauLifshitz equation with a strong easyaxis anisotropy.Revista Matemática Iberoamericana3712021, 95128
 17 articleRecent results for the LandauLifshitz equation.SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada2021
 18 articleModerate Deviations for the SSEP with a Slow Bond.Journal of Statistical PhysicsFebruary 2021
 19 articleNonequilibrium Fluctuations of the Weakly Asymmetric Normalized Binary Contact Path Process.Stochastic Processes and their Applications2021
Reports & preprints
 20 miscStochastic modulational instability in the nonlinear Schrödinger equation with colored random dispersion.November 2021
 21 miscInitial value problem for onedimensional rotating shallow water equations.March 2021
 22 miscComparison principles and applications to mathematical modelling of vegetal metacommunities.November 2021
 23 miscHigh order linearly implicit methods for evolution equations: How to solve an ODE by inverting only linear systems.November 2021
 24 miscOn the hydrodynamics of active matter models on a lattice.November 2021
 25 miscAsymmetric attractive zerorange processes with particle destruction at the origin.November 2021
 26 miscExistence and decay of traveling waves for the nonlocal GrossPitaevskii equation.November 2021
 27 miscSteady state large deviations for onedimensional, symmetric exclusion processes in weak contact with reservoirs.November 2021
8.3 Cited publications

28
articleGlobal Schrödinger maps in dimensions
$d2$ : small data in the critical Sobolev spaces.Annals of Mathematics2011, 14431506  29 unpublishedExponential integrators for the stochastic Manakov equation.May 2020, https://arxiv.org/abs/2005.04978  working paper or preprint
 30 articleMacroscopic fluctuation theory.Reviews of Modern Physics872Jun 2015, 593–636URL: http://dx.doi.org/10.1103/RevModPhys.87.593
 31 articleHydrodynamic limit for a facilitated exclusion process.Annales de l'IHP  Probabilités et Statistiques5612020, 667714
 32 articleJournal of Statistical Physics1073/42002, 599–634URL: http://dx.doi.org/10.1023/A:1014555927320
 33 articleCoercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker Planck equations.Numerische Mathematik144https://arxiv.org/abs/1802.02173v12020
 34 unpublishedSuperdiffusion of energy in a chain of harmonic oscillators with noise.2014, working paper or preprint

35
articleOn Schrödinger maps from
${T}^{1}$ to${S}^{2}$ .Ann. Sci. ENS452012, 637680  36 articleExact Hydrodynamic description of active lattice gases.Physics Review Letter1202680032018
 37 articleThe sineGordon regime of the LandauLifshitz equation with a strong easyplane anisotropy.Ann. Inst. H. Poincaré Anal. Non Linéaire3572018, 18851945URL: https://doi.org/10.1016/j.anihpc.2018.03.005
 38 articleTraveling waves for some nonlocal 1D GrossPitaevskii equations with nonzero conditions at infinity.Discrete Contin. Dyn. Syst.4012020, 635682URL: https://doi.org/10.3934/dcds.2020026
 39 bookMicromagnetics and Recording Materials.http://dx.doi.org/10.1007/9783642285776SpringerVerlag Berlin Heidelberg2012, URL: https://doi.org/10.1007/9783642285776
 40 articleHydrodynamics of the weakly asymmetric normalized binary contact path process.Stochastic Processes and their Applications130112020, 67576782
 41 articleTopological computation based on direct magnetic logic communication.Scientific Reports52015, URL: http://dx.doi.org/10.1038/srep15773