• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

After recalling the remarkable formulation made in 1926 by Erwin Madelung of the Schrödinger equation in terms of fluid mechanics, I will introduce a rational scheme, based on the least action principle and some non-linear rescaling of the time variable, starting from Euler's equations of isothermal compressible fluids (1755), followed by Fourier's heat conduction equation (1807), leading to Schrödinger's equation of quantum mechanics (1925). Finally, I will suggest the application of this scheme to Magneto-hydrodynamics. Madelung, E. (1926). "Eine anschauliche Deutung der Gleichung von Schrödinger". Naturwissenschaften 14 (45): 1004–1004.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

The kinetic wave equation is of central importance in the theory of weak turbulence, but no rigorous derivation of it is known. I will show how it can be derived from NLS on the torus with random forcing, in the small nonlinearity / big box limit. This is joint work with Isabelle Gallagher and Zaher Hani.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

This talk proposes a quantitative convergence estimate for the mean-field limit of the N-body Schrödinger equation that is uniform in the classical limit. It is based on a new variant of the Dobrushin approach for the mean field limit in classical mechanics, which avoids the use of particle trajectories and empirical measures, and has a very natural quantum analogue. (Work in collaboration with C. Mouhot and T. Paul).

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

After recalling old or more recent point of views on bosonic quantum field theory and mean field problems, the series of works in collaboration with Z. Ammari will be summarized. This phase-space presentation implements the old dream of an infinite dimensional microlocal analysis. In particular the mean field dynamics is nothing but a propagation of singularity result in the semiclassical regime. This talk will put the stress on the key issues related with the infinite dimensional setting and on the new results for the mean field problem provided by this approach.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

In my thesis, jointly supervised by N.J. Mauser and F. Nier, we deal with approximations of the time-dependent linear many body Schrödinger equation with a two particles interaction potential, by introducing a discrete version of the equation and mean field limits. We consider the bosonic Fock space in a finite dimensional setting. Mathematical tools include the reduced density matrices and Wigner measure techniques exploiting the formal analogy to semi-classical limits.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

"The talk is planned to revisit Grenier's scheme for instability of Euler and Prandtl, introduced in his CPAM-2000 paper, and to present some new applications in the instability of generic boundary layers and instability of Vlasov-Maxwell in the classical limit".

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

We use quantum Rényi divergences to define "correlation" functionals of many-fermion states (density operators on a Fock space). The "reference" state for the relative entropy functional is the unique gauge-invariant quasi-free (g.i.q.f.) state with the same 1-RDM as the state of interest. That is, the "correlation" of the state of interest is its Rényi divergence from the uniquely associated g.i.q.f. state. Correlation functionals defined in this way enjoy the following properties: (a) they take only non-negative values, possibly infinity; (b) they assign the value 0 to all Slater determinant states; (c) they are monotone with respect to restriction of states; (d) they are additive over independent subsystems; and (e) they are invariant under changes of the 1-particle basis (Bogoliubov transformations). The quantum relative entropy or quantum Kullback-Leibler divergence is a special and distinguished member of any family of quantum Rényi divergences (of which there are at least two). The associated correlation functional, defined using quantum Kullback-Leibler divergence, we call "nonfreeness." Nonfreeness enjoys further appealing properties not shared by related correlation functionals: (f) the nonfreeness of a state X is the minimum possible value for the entropy of X relative to any g.i.q.f. reference state; (g) there is a simple formula for a pure state's nonfreeness in terms of it's natural occupation numbers; and (h) within the convex set of n-fermion states with given 1-RDM, the nonfreeness minimizer equals the entropy maximizer, which is the Gibbs canonical (n-particle) state.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

I report on current work with Toan Nguyen and Francois Golse.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

In this talk, I will present various time integrators for NLS equations when the potentials are time dependent. In this case, the usual time splitting schemes fail. I will introduce exponential Runge-Kutta scheme and Lawson scheme and present some of their properties.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

In this talk, I investigate the error behaviour of exponential operator splitting methods for nonlinear evolutionary problems. In particular, I will present an exact local error representation that is suitable in the presence of critical parameters. Essential tools in the theoretical analysis including time-dependent nonlinear Schrödinger equations in the semi-classical regime as well as parabolic initial-boundary value problems with high spatial gradients are an abstract formulation of differential equations on function spaces and the formal calculus of Lie-derivatives.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

We introduce accurate and efficient methods for nonlocal potentials evaluations with free boundary condition, including the 3D/2D Coulomb, 2D Poisson and 3D dipole-dipole potentials. Both methods rely on the same assumption: the density is smooth and fast decaying. The first method,proposed by Jiang, Greengard and Bao, evaluates the potential in spherical/polar coordinates using NonUniform FFT algorithm, where the singularity of the Fourier representation disappears automatically, while the second one is based on a Gaussian-sum approximation of the singular convolution kernel and Taylor expansion of the density. Both methods are accelerated by fast Fourier transforms (FFT). They are accurate (14-16 digits), efficient ($O(Nlog N)$ complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelizable.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

The perfectly matched layers (PML) and exterior complex scaling (ECS) methods for absorbing boundary conditions are analyzed using spectral decomposition. Both methods are derived as analytical continuations of unitary to contractive transformations. We find that the methods are mathematically and numerically distinct: ECS is complex stretching that rotates the operator's spectrum into the complex plane, whereas PML is a complex gauge transform which shifts the spectrum. Consequently, the schemes differ in their time-stability. Numerical examples are given.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Summer school "Around Schrödinger equations" (2015)

In this talk, we investigate the properties of solitonic structures arising in quadratic media. More precisely, we look for stationary states in the context of normal or anomalous dispersion regimes, that lead us to either elliptic or nonelliptic systems and we address the problem of orbital stability. Finally, we present some numerical experiments in order to compute localized states for several regimes.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We will survey recent results and open problems on various nonlocal "full dispersion" models of surface water waves.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

The usual model of nonlinear optics given by the cubic NLS equation is too crude to describe large intensity phenomenas such as filamentation, which modifies the focusing of laser beams. I shall explain how to derive some more appropriate variants of the NLS model from Maxwell's equations, using improved approximations of the original dispersion relation or taking ionization effects into account. I shall provide rigorous error estimates for the models considered, and also discuss some open problems related to these modified NLS equations. This is joint work with David Lannes and Jeremie Szeftel.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation with a critical nonlinearity and prove the asymptotic stability of solutions in the neighborhood of the self similar solutions under the non zero mass condition and the smallness on the data.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin as a model of laser-plasma interaction. In this talk, we prove the well-posedness of this system for the periodic initial data. In particular, if the coefficients of Laplacian satisfy some conditions, then the well-posedness is proved at the scaling critical regularity by using U^2 and V^2 spaces.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider the 1D Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we study existence and stability of the black solitons with a combination of variational and perturbation arguments. This is a joint work with Isabella Ianni and Julien Royer.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

In this talk, I will explain the Proudman resonance. It is a resonant respond in shallow waters of a water body on a traveling atmospheric disturbance when the speed of the disturbance is close to the typical water wave velocity. In order to explain this phenomenon, I will prove a local well-posedness of the water waves equations with a non constant pressure at the surface, taking into account the dependence of small physical parameters. Then, I will justify mathematically the historical work of Proudman. Finally, I will study the linear water waves equations and I will give dispersion estimates in order to extend The Proudman resonance to deeper waters. To complete these asymptotic models, I will show some numerical simulations.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider a system of nonlinear Schrodinger equations with cubic nonlinearity, called a coherently coupled NLS system (CCNLS) in nonlinear optics, in one space dimension. We study orbital stability and instability of standing wave solutions of (CCNLS), and prove similar results to Colin and Ohta (2012) which studies a system of NLS equations with quadratic nonlinearity. This is a joint work with Shotaro Kawahara (Tokyo University of Science).

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

The equation in (BI) appears for instance in the Born-Infeld nonlinear electromagnetic theory: in the electrostatic case it corresponds to the Gauss law in the classical Maxwell theory and so  is the electric potential and  is an assigned extended charge density. We discuss existence, uniqueness and regularity of the solution of (BI). The results have been obtained in a joint work with Denis Bonheure and Pietro d’Avenia.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

A new NLS type equation is employed for modeling long-range interactions in nonlinear optics, in a collaboration with experimental physicists. It is of quasilinear type and models fluctuations around a 'continuous-wave polariton' which are chosen according to Bogoliubov theory. We present a numerical discretization method and simulation results. Mathematical theory for this equation is work in progress.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

In this talk, we consider a quasiinear Schrodinger equation which appears in the study of plasma physics. We are interested in the uniqueness of ground states without assuming any restriction on a physical parameter. We also study asymptotic behavior of ground states as the parameter goes to zero.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We show that space-time resonances induce high-frequency instabilities in the two-fluid Euler-Maxwell system. This implies in particular that the Zakharov approximation to Euler-Maxwell is stable if and only if the group velocity vanishes. The instability proof relies on a short-time representation formula for the flows of pseudo-differential operators of order zero. This is joint work with Eric Dumas (Grenoble) and Lu Yong (Prague).

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)