Recherche

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 28, 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.

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 20 and 19, A. de Laire, P. Gravejat and D. Smets 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.

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 and G. Ferriere study this equation, named the logarithmic Gross-Pitaevskii equation (or logGP), on the whole space ${ℝ}^d$ in 12. 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…

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 22, Q. Chauleur and A. Mouzard 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.

Asymmetric attractive zero-range process with particle destrucsion at the origin

In 14, C. Erignoux, M. Simon and L. Zhao investigate the macroscopic behavior of asymmetric attractive zero-range processes on $\mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N\beta$ , where $\beta \in ℝ$ and $N\in \mathbb{N}$ is the scaling parameter. They prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of $\beta$ . Namely, if $\beta \ge 0$ , then the boundary condition prescribes the particle current through the origin, whereas if $\beta <0$ , the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.

Large deviations principle for the SSEP with weak boundary interactions

Efficiently characterizing non-equilibrium stationary states (NESS) has been in recent years a central question in statistical physics. The Macroscopic Fluctuations Theory 31 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 11, 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 37, where the SSEP with strong boundary interactions was considered. In 11, 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 non-linear differential equation. The obtained differential equation is the same as in 37, with different boundary conditions corresponding to the different scales of boundary interaction.

Mapping hydrodynamics for the facilitated exclusion and zero-range processes

In 15, C. Erignoux, M. Simon and L. Zhao derive the hydrodynamic limit for two degenerate lattice gases, the facilitated exclusion process (FEP) and the facilitated zero-range 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 they extensively use to derive its hydrodynamic limits in both cases, the FEP is not. To derive the hydrodynamic limit for the latter, they exploit that of the zero-range process, together with a classical mapping between exclusion and zero-range processes, both at the microscopic and macroscopic level. Because the FEP is degenerate, we had to develop new mapping tools to prove hydrodynamic in the asymmetric case. In the symmetric case, a proof already existed 33 for the hydrodynamic limit, however our mapping arguments further provide an alternative, simpler proof.

Stationary fluctuations for the facilitated exclusion process

In 27, C. Erignoux and L. Zhao derive the stationary fluctuations for the Facilitated Exclusion Process (FEP) in one dimension in the symmetric, weakly asymmetric and asymmetric cases. The proof relies on the mapping between the FEP and the zero-range process, and extends the same strategy as in previous works, where hydrodynamic limits were derived for the FEP, to its stationary fluctuations. Their results thus exploit works on the zero-range process’s fluctuations, but they also provide a direct proof in the symmetric case, for which they derive a sharp estimate on the equivalence of ensembles for the FEP’s stationary states.

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 13 S. De Bièvre provides 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 nonclassicality 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. S. De Bièvre establishes sharp bounds of very general validity that imply that the support uncertainty is an efficient Kirkwood-Dirac nonclassicality witness for pure states. When adapted to completely incompatible observables that are close to mutually unbiased ones, this bound allows to fully characterize the Kirkwood-Dirac classical pure states as the eigenvectors of the two observables. In 29, De Bièvre, C. Langrenez and D. Arvidsson (Cambridge) provide an analysis of the geometry of the KD-positive and -nonpositive pure and mixed states. They characterize the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B.

Photon-added/subtracted states: nonclassicality

Photon addition and subtraction render Gaussian states of the quantized optical field non-Gaussian. In 17, S. De Bièvre and A. Hertz (Toronto-Ottawa) provide a quantitative analysis of the change in the so-called nonclassicality produced by these processes by analyzing the Wigner negativity and quadrature coherence scale (QCS) of the resulting states. The QCS is a recently introduced measure of nonclassicality [PRL 122, 080402 (2019), PRL 124, 090402 (2020)], that we show to undergo a relative increase under photon addition/subtraction that can be as large as 200%.

Interferometric measurement of the QCS

In 16, S. De Bièvre and his collaborators from the Université Libre de Bruxelles provided an experimental procedure for directly accessing the QCS of the quantum state of an optical field, with the help of only a simple linear interferometer involving two replicas (independent and identical copies) of the state $\widehat{\rho }$ supplemented with photon-number-resolving measurements. The proposed protocol has since been implemented with success on the cloud quantum computer of Xanadu, by a team of physicists from the Universities of Toronto and Ottawa.

Modulational

In 9 S. De Bièvre, G. Dujardin and their collaborators (physicists from the PhLAM laboratory in Lille) study modulational instability in a dispersion-managed system where the sign of the group-velocity dispersion is changed at uniformly distributed random distances around a reference length. An analytical technique is presented to estimate the instability gain from the linearized nonlinear Schrödinger equation, which is also solved numerically. The comparison of numerical and analytical results confirms the validity of their approach. Modulational instability of purely stochastic origin appears.

Approach to equilibrium in quantum systems

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 23, 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 21. 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.

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 has been established recently by A. de Laire and S. López-Martínez in 8. Mathematically, these solitons correspond to minimizers of the energy at fixed momentum and are orbitally stable. In the paper 18, A. de Laire, G. Dujardin and S. López-Martínez 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.

Linearly implicit high-order numerical methods for evolution problems

G. Dujardin and his collaborator 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 his collaborator 24. 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 25.

Exponential integrators for the stochastic Manakov system

The Manakov system is a system of dispersive stochastic PDEs modelling the propagation of light in optical fibers taking into account the polarization mode dispersion effects. In 10, G. Dujardin and his collaborators developed and analyzed an exponential integrator for the numerical solution of this stochastic PDE system. In particular, they proved that this exponential integrator has strong order $1/2$ for a truncated nonlinearity and they infered that is also has order $1/2$ in probability and order $1/2$ almost surely, for general nonlinearities. Moreover, they provided several numerical experiments illustrating their theoretical results as well as the efficiency of their numerical integrator.

Uniform estimates for numerical schemes applied to parabolic problems with Neumann boundary conditions

In 26, G. Dujardin and his collaborator tackled the problem of proving uniform-in-time order estimates for a scheme integrating the linear heat equation with homogeneous pure Neumann boundary conditions on a bounded interval. Despite the lack of consistency of the discretization of the boudary condition with the Laplace operator, they proved that the scheme they consider is of order 1 in space and time uniformly-in-time. They applied this result to the question of the numerical computation of stationary states to nonhomogeneous heat equations.