Research

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

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

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

Logarithmic Gross-Pitaevskii equation

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

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

In the preprint 36, G. Ferriere examines the Landau-Lifshitz-Gilbert (LLG) equation governing the magnetization dynamics in an infinite ferromagnetic nanowire with easy-axis anisotropy along the $e_1$ direction and subjected to a constant external magnetic field $h_0{e}_1$. Under specific conditions on $h_0$, the study establishes the existence of stationary solutions with identical asymptotic behavior at infinity, their uniqueness up to the symmetries of the LLG equation, and the instability of their orbits under the LLG flow. These findings provide new insights into the behavior of solutions to the LLG equation, complemented by numerical simulations that explore the stability of 2-domain wall structures and the interactions between domain walls.

Existence and Uniqueness of Domain Walls for Notched Ferromagnetic Nanowires

In the preprint 34, R. Côte (University of Strasbourg), C. Courtès (University of Strasbourg), G. Ferriere, L. Godard-Cadillac (University of Bordeaux), and Y. Privat (University of Lorraine, Nancy) explore the existence and properties of domain walls in a model of notched ferromagnetic nanowires. They employ variational methods and critical point theory to investigate the energy functional describing the system.

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

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

Numerical computation of dark solitons of a nonlocal nonlinear Schrödinger equation

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

Gray and black solitons of nonlocal Gross-Pitaevskii equations

In the preprint 39, A. de Laire and S. López-Martínez (Autonomous University of Madrid) continue the investigation started in 14, 25, concerning the qualitative aspects of dark solitons of one-dimensional Gross-Pitaevskii equations with general nonlocal interactions. Under general conditions on the potential interaction term, they provide uniform bounds, demonstrate the existence of symmetric solitons, and identify conditions under which monotonicity is lost. Additionally, they present new properties of black solitons. Moreover, they establish the nonlocal-to-local convergence, i.e. the convergence of the soliton of the nonlocal model toward the explicit dark solitons of the local Gross-Pitaevskii equation.

Exotic traveling waves for a quasilinear Schrödinger equation with nonzero background

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

In the preprint 38, they investigate how the quasilinear term affects the traveling-waves solutions. They provide a complete classification of finite energy traveling waves of the equation, in terms of the two parameters: the speed and the strength of the quasilinear term. This classification leads to the existence of dark and antidark solitons, as well as more exotic localized solutions like dark cuspons, compactons, and composite waves, even for supersonic speeds. Depending on the parameters, these types of solutions can coexist, showing that finite energy solutions are not unique. Furthermore, they prove that some of these dark solitons can be obtained as minimizers of the energy, at fixed momentum, and that they are orbitally stable.

Traveling waves for a quasilinear Schrödinger equation

In the paper 42, E. Le Quiniou studies a quasilinear Schrödinger equation with nonzero conditions at infinity. In the previous work 38 with A. de Laire, he obtained a continuous branch of traveling waves, given by dark solitons indexed by their speed. Neglecting the quasilinear term, one recovers the Gross–Pitaevskii equation, for which the branch of dark solitons is stable. It is known that the Vakhitov–Kolokolov (VK) stability criterion or momentum of stability criterion holds for general semilinear equations with nonvanishing conditions at infinity. In the quasilinear case, E. Le Quiniou proves that the VK stability criterion still applies and he deduces that the branch of dark solitons is stable for weak quasilinear interactions. For stronger quasilinear interactions, a cusp appears in the energy-momentum diagram, implying the stability of fast waves and the instability of slow waves.

Travelling waves for the Gross–Pitaevskii equation on the strip

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

In the articles 27 and 26, A. de Laire, P. Gravejat (CY Cergy Paris University) and D. Smets (Sorbonne University) provide a rigorous framework for studying this kind of phenomenon. Precisely, they prove the existence of nonconstant finite energy travelling wave solutions to the Gross-Pitaevskii equation on the strip $ℝ\times {𝕋}_L$, obtained as minimizers of the energy at fixed momentum. Moreover, by studying the associated variational problem, they deduce that these minimizers are exactly the planar dark solitons when $L$ is less than a critical value, and that they are genuinely two-dimensional solutions otherwise. In particular, planar solitons do not minimize the energy in the presence of a large transverse direction. The proof of the existence of minimizers is based on the compactness of minimizing sequences, relying on a new symmetrization argument that is well-suited to the periodic setting.

Standing wave for two-dimensional Schrödinger equations with discontinuous dispersion

In collaboration with B. Alouini (University of Monastir) and I. Manoubi (Université of Gabès) 30, O. Goubet has studied the existence and stability of standing wave for an evolution nonlinear Schrödinger equation with discontinuous dispersion, the discontinuity being supported by a straight line. Both pure power nonlinearities and logarithmic nonlinearities are considered. The discontinuity destroys the invariance by space translation for the equation. The main result is that when restricted to a suitable subspace that contains the standing waves, these waves are orbitally stable in the $H^1$ subcritical regime in the pure power case or in the logarithmic case, and strongly unstable in the critical or supercritical case.

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

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

Non-Linear Problems in Interacting Particle Systems

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

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

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

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

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

Kirkwood-Dirac distributions

The Kirkwood-Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables A and B. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In 50, S. De Bièvre provided an in-depth study of the notion of completely incompatible observables that he recently introduced and of its links to the support uncertainty and to the Kirkwood-Dirac nonpositivity of pure quantum states. The latter notion has recently been proven central to a number of issues in quantum information theory and quantum metrology. In this last context, it was shown that a quantum advantage requires the use of Kirkwood-Dirac nonclassical states.

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

Characterizing the geometry of the Kirkwood-Dirac positive states

In 28, S. De Bièvre, C. Langrenez and D. R. M. Arvidsson-Shukur (Hitachi Cambridge Laboratory) analyse the geometry of the KD-positive and -nonpositive pure and mixed states. They analyze the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B and provide an algebraic necessary and sufficient condition for this set to be minimal, meaning that it contains only the basis projectors of A and B.

Convex roofs witnessing Kirkwood-Dirac nonpositivity

In 40, S. De Bièvre, C. Langrenez and D. R. M. Arvidsson-Shukur (Hitachi Cambridge Laboratory) introduce and study two witnesses for KD nonpositivity, through a convex roof construction and the notion of support uncertainty.

The set of Kirkwood-Dirac positive states is almost always minimal

In 41, S. De Bièvre, C. Langrenez and their collaborators show that, with probability one with respect to the choices of A and of B, the set of KD-positive states is indeed minimal, in the above sense. They also provide examples where this set is not minimal and contains “exotic” KD-positive states, which are mixed but cannot be written as mixtures of pure KD-positive states.

Contextuality Can be Verified with Noncontextual Experiments

In 43, S. De Bièvre and his collaborators show how the exotic states introduced in 41 can be used to evidence contextuality with a noncontextual experiment.

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

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

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

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

Strong convergence for the discrete nonlinear Klein-Gordon equation

In 31, Q. Chauleur extends the analysis of nonlinear dispersive equations, such as the nonlinear Klein-Gordon equation, on an infinite lattice $h{\mathbb{Z}}^d$ as the lattice spacing $h\to 0$ approaches the continuum limit. This work builds upon the framework established in 18, employing bilinear estimates of the Shannon interpolation combined with controls on the growth of discrete Sobolev norms of the solutions. Additionally, Q. Chauleur also provides some perspectives on uniform dispersive estimates for nonlinear waves on lattices.

Linearly implicit high-order numerical methods for evolution problems

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

Vortex nucleation in 2D rotating Bose–Einstein condensates

In 24, G. Dujardin, I. Lacroix-Violet (University of Lorraine, Nancy) and A. Nahas (University of Lille) introduce a new numerical method for the minimization under constraints of a discrete energy modeling multicomponents rotating Bose-Einstein condensates in the regime of strong confinement and with rotation. Moreover, they consider both segregation and coexistence regimes between the components. It is well known that, depending on the regime, the minimizers may display different structures, sometimes with vorticity (from singly quantized vortices, to vortex sheets and giant holes). In order to study numerically the structures of the minimizers, the authors of 24 introduce a numerical algorithm for the computation of the indices of the vortices, as well as an algorithm for the computation of the indices of vortex sheets. Several computations are carried out, to illustrate the efficiency of the method, to cover different physical cases, to validate recent theoretical results as well as to support conjectures. Moreover, the new methods is compared with an alternative method from the literature. This work was part of A. Nahas’ PhD thesis, co-advised by I. Lacroix-Violet (University of Lorraine, Nancy) and G. Dujardin.

Finite volumes for the Gross-Pitaevskii equation

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

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

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

Discrete quantum harmonic oscillator and Kravchuk transform

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

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

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