We survey new and old results on the Intermediate Long Wave (ILW) equation from modeling, PDE and integrability aspects.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

For solutions of the Benjamin-Ono equation with periodic boundary conditions, I will discuss the link in the high frequency regime between the nonlinear Fourier transform inherited from the integrable structure, and a gauge transform introduced by T. Tao in 2004 in the context of the low regularity initial value problem. As an application, we will get optimal high frequency approximations of solutions. This talk is based on a recent joint work with T. Kappeler and P. Topalov.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

Using the Birkhoff map of the Benjamin-Ono equation as a starting point, we deform it near an arbitrary compact family of finite dimensional tori, invariant under the Benjamin-Ono flow, so that the following main properties hold: (i) When restricted to the family of finite dimensional tori, the transformation coincides with the Birkhoff map. (ii) Up to a remainder term, which is smoothing to any given order, it is a pseudo-differential operator of order 0, with principal part given by the Fourier transform, modified by a phase factor. (iii) The transformation is canonical and the pullback of the Benjamin-Ono Hamiltonian by it is in normal form up to order three. Such coordinates are a key ingredient for studying the stability of finite gap solutions of arbitrary size of the Benjamin-Ono equation under small, quasi-linear, momentum preserving perturbations. This is joint work with Riccardo Montalto.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

We present a detailed numerical study of solutions to Davey-Stewartson (DS) II systems, nonlocal non-linear Schrödinger equations in two spatial dimensions. A possible blow-up of solutions is studied, a conjecture for a self-similar blow-up is formulated. In the integrable cases, numerical and hybrid approaches for the inverse scattering are presented.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

Currently, the studies on periodic travelling waves of the nonlinear dispersive equations are becoming very popular. In this study we investigate the spectral stability of the periodic waves for the fractional Benjamin-Bona-Mahony (fBBM) equation, numerically. For the numerical generation of periodic travelling wave solutions we use an iteration method which is based on a modification of Petviashvili algorithm. This is a joint work with S. Amaral, H. Borluk, G.M. Muslu and F. Natali.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

We are concerned with finding Fokker-Planck equations in whole space with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrary close to its infimum. This infimum is 1, corresponding to the high-rotational limit in the Fokker-Planck drift. Such an optimal Fokker-planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoercive. The proof is based on the recent result that the L2- projector norms of the Fokker-Planck equation and of its drift-ODE coincide. Finally we give an outlook onto using Fokker-Planck equation with t-dependent coefficients. This talk is based on a joint work with Beatrice Signorello.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

An efficient high precision hybrid numerical approach for integrable Davey-Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study of DS I solutions in this class. Localized stationary solutions are constructed and shown to be unstable against dispersion and blow-up. A finite-time blow-up of initial data in the Schwartz class of smooth rapidly decreasing functions is discussed.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

The majority of dispersive equations in one space-dimension can be realized as dispersive perturbations of the Burgers equation ut + uux = Lux, where L is a local or nonlocal symmetric operator. For negative order dispersion, the Burg- ers’ nonlinearity dominates and classical solutions break down due to shock-formation/wave- breaking. Using hyperbolic techniques we establish global existence and uniqueness of entropy solutions, with L2 initial data, for a family of negative order dispersive equations, but our main focus will be on a new generalization of the classical Oleïnik estimate for Burgers equation. We obtain one sided Hölder regularity for the solutions, which in turn controls their height and provides a novel bound of the lifespan of classical solutions based on their initial skewness. This is joint work with Jun Xue (NTNU).

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

We study the unconditional uniqueness of solutions to the Benjamin-Ono equation with initial data in Hs, both on the real line and on the torus. We use the gauge transformation of Tao and two iterations of normal form reductions via integration by parts in time. By employing a refined Strichartz estimate we establish the result below the regularity threshold s = 1/6. As a by-product of our proof, we also obtain a nonlinear smoothing property on the gauge variable at the same level of regularity. This talk is based on a joint work with Razvan Mosincat (University of Bergen).

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Dispersive equations and systems: IST and PDE methods" (2021)

This talk presents a joint mean-field and classical limit by which the Euler-Poisson system is rigorously derived from the N-body Schrödinger equation with Coulomb interaction in space dimension 3. One of the key ingredients in this derivation is a remarkable inequality for the Coulomb potential which has been obtained by S. Serfaty in 2020 (Duke Math. J.). 2)

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)

The Pauli-Poisson equation is a semi-relativistic description of electrons under the influence of a given linear (strong) magnetic field and a self-consistent electric potential computed from the Poisson equation in 3 space dimensions. It is a system of two magnetic Schrödinger type equations for the two components of the spinor, coupled by the additional Stern-Gerlach term of magnetic field and spin represented by the Pauli matrices. On the other hand the Pauli-Poiswell equation includes the self-consistent description of the magnetic field by coupling it via a three Poisson equations with the Pauli current as source term to the Pauli equation. The Pauli-Poiswell equation offers a fully self-consistent description of spin-1/2-particles in the semi-relativistic regime. We introduce the equations and study the semiclassical limit of Pauli-Poisson towards a semi-relativistic Vlasov equation with Lorentz force coupled to the Poisson equation. We use Wigner transform methods and a mixed state formulation, extending the work of Lions-Paul and Markowich-Mauser on the semiclassical limit of the Schrödinger-Poisson equation. We also present a result on global weak solutions of the Pauli-Poiswell equation.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)

Based on a series of works in collaboration with Gilles Lebeau and Nicolas Burq -Propagation of smallness and control for heat equations (with Nicolas Burq, to appear in JEMS), -Spectral Inequalities for the Schrödinger operator (with Gilles Lebeau). -Propagation of smallness and spectral estimates (with Nicolas Burq) And the recent advances in propagation of smallness introduced by Logonuv and Malinnikova. A. Logunov and E. Malinnikova. Quantitative propagation of smallness for solutions of elliptic equations. Preprint, Arxiv, (arXiv:1711.10076), 2017 A. Logunov. Nodal sets of Laplace eigenfunctions : polynomial upper estimates of the Hausdorff measure. Ann. of Math. (2), 187(1):221–239, 2018.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)

The aim of quasilinear theory is to explain relaxation or saturation of kinetic instabilities governed by the Vlasov-Poisson (VP) equation, by showing that in fact the Hamiltonian dynamics of VP can be approximated by a diffusion equation in velocity for the space-average distribution function.

• Thematic program: Quantum Equations and Experiments (2021/2022)
• Event: Workshop on "Kinetic theory: Boltzmann, Fokker Planck - Balescu, Lenhard” (2021)