This is a joint work with Christophe Cheverry. We study high frequency so- lutions of nonlinear hyperbolic equations for time scales at which dispersive and nonlinear effects can be present in the leading term of the solution, on a model stemming from strongly magnetized plasmas or nuclear magnetic reso- nance experiments. We show how the produced waves can accumulate during long times to produce constructive and destructive interferences which, in the above contexts, are part of turbulent effects.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

In this work we will look at the focusing Davey-Stewartson equation from two different angles, using advanced numerical tools. As a nonlinear dispersive PDE and a generalisation of the non-linear Schr¨odinger equation, DS possesses solutions that develop a singularity in finite time. We numerically study the long time behaviour and potential blow-up of solutions to the focusing Davey-Stewartson II equation for various initial data and propose a conjecture describing the blow up rate and solution profiles near the singularity. Secondly, DS is an integrable system and can be studied as an inverse scat- tering problem. Both the forward and inverse scattering transformation in this case are reduced to a d-bar system which plays the role that Riemann-Hilbert problems play in one dimensional problems. We will present numerical solutions for Schwartzian and compactly supported potentials. Further, to complement numerics, we will discuss analytical considerations to handle asymptotic be- haviour. In all studied cases we use spectral methods and achieve machine pre- cision. Based on joint works with Christian Klein and Ken McLaughlin

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

We discuss numerical methods to construct solutions to nonlinear dispersive PDEs on the whole real line, and this for initial data which are slowly decreasing towards infinity or just bounded there. As an example we discuss the transverse stability of the Peregrine solution in the 2d nonlinear Schrodinger equation.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

This is a jointwork with Thomas Kappeler. Using the Lax pair structure for the Benjamin-Ono equation with periodic boundary conditions, we construct a global system of Birkhoff coordinates on the phase space of real valued square integrable functions with average 0 on the torus, including a characterisation of finite gap potentials. Among consequences, we infer almost periodicity of all trajectories, identification of traveling waves and construction of periodic in time solutions with low regularity.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The Serre-Green-Naghdi system is a fully nonlinear and weakly dispersive model for the propagation of surface gravity waves. It enjoys many good theoretical properties, including a robust well-posedness theory for the initial-value prob- lem, and a Hamiltonian structure. It is however not so suitable for practical use, as standard numerical strategies involve the costly inversion of an elliptic operator at each time step. N. Favrie and S. Gavrilyuk proposed a novel strat- egy for efficiently producing approximate solutions, by introducing a “relaxed” first-order quasilinear system of balance laws, depending on additional unknows and a free parameter. The claim is that in the singular limit when the param- eter goes to infinity, solutions of the relaxed system approach solutions of the Serre-Green-Naghdi system. We will discuss a rigorous analysis. It differs from standard results due to the presence of an additional parameter (describing the shallowness of the flow) and order-zero source terms which become dominant when the shallowness parameter goes to zero.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The linear stability of homogenous shear flows between two rigid walls is a clas- sical problem that goes back to Rayleigh (1880). Among other things, Rayleigh was able to show that a shear flow with no inflection points is linearly stable. The generalisation of this stability criterion to the free-surface setting is not straightforward and was established much later by Yih (1971) (under certain restrictions) and, more recently, Hur & Lin (2008). In the case when a shear flow with a free surface is modelled by constant vorticity layers, no stability criterion is known. As a first step in this direction we consider the stability analysis of a bilinear shear current and establish a criterion for the stability of the flow. The effect of density stratification on the stability of the flow will also be investigated.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The binormal flow is a model for the dynamics of a vortex filament in a 3-D in- viscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schr¨odinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results presented in this talk is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conservation law in the framework of fluid mechanics, it involves the amplitude of the Fourier modes of the variation of the direction of the vorticity. This is a joint work with Luis Vega.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

In a series of papers with Sylvie Benzoni-Gavage (and, depending on papers, Pascal Noble or Colin Mietka), we have studied both co-periodic stability and modulation systems for periodic traveling waves of a rather large class of Hamil- tonian partial differential equations that includes quasilinear generalizations of the Korteweg–de Vries equation and dispersive perturbations of the Euler equa- tions for compressible fluids, either in Lagrangian or in Eulerian coordinates. All characterizations are derived in terms of the Hessian matrix of the action integral of profile equations, a finite-dimensional object. In the present talk, with this in mind, we shall discuss the consequences of the recently obtained expansions of this matrix in two asymptotic regimes, namely the zero-amplitude and the zero-wavelength limits.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

In this presentation, we consider the nonlinear Schr¨odinger equation with loga- rithmic nonlinearity (logNLS in short). We mostly focus on the focusing case which presents a very special Gaussian stationary solution, called Gausson, which is orbitally stable. In fact, more generally, it has been shown that every Gaussian data remains Gaussian through the flow of logNLS, and this feature gives rise to (almost) periodic solutions in the focusing case, called breathers. The main result of this talk addresses the existence of multi-solitons, i.e. solu- tions to logNLS which behaves like the sum of several solitons (i.e. Gaussons here) for large times, in dimension 1. This kind of result is rather usual for dispersive equations with polynomial-like nonlinearity, and our proof is directly inspired from the usual proof with energy techniques. The main difficulty is the fact that the energy cannot be linearized as one would want, at least not ev- erywhere. Furthermore, some new and surprising features appear in this result: the convergence is in H1 and (H1) with a rate faster than exponential, and there is no need for a large enough relative speed (non-zero is sufficient).

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

The Euler-Korteweg system is a dispersive perturbation of the usual compress- ible Euler equations that includes the effect of capillary forces. For small ir- rotational initial data, global well-posedness is known to hold in dimension at least three. In this talk we discuss the case of small initial data with non zero vorticity, where the dispersive system becomes a coupled dispersive-transport system. The main result is that the time of existence only depends on the size of the initial vorticity.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)

This lecture is devoted to the study of the Hele-Shaw equation, based on a joint work with Nicolas Meunier and Didier Smets. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancel- lations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equa- tions in a very simple way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapounov functions of different natures. We also deduce from these identities and previous works on the water wave problem a simple proof of the well-posedness of the Cauchy problem.

• Thematic program: Quantum: Equations and Experiments (2019/2020)
• Event: Workshop on "Some news on Dispersive PDEs: Modeling, Theory and Numerics" (2019)