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)