In this talk I shall present some results on global existence, scattering and blow-up for the focusing nonlinear Schrödinger equation on hyperbolic space. This is a joint work with Thomas Duyckaerts.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

We consider the two dimensional focusing cubic nonlinear Schrodinger equation. Bourgain and Wang have constructed smooth solutions which blow up in finite time with the pseudo conformal speed, and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynamic is unstable. More precisely, we show that any such solution with small super critical L^2 mass lies on the boundary of both H^1 open sets of global solutions that scatter forward and backwards in time, and solutions that blow up in finite time on the right in the log-log regime. This is a joint work with F. Merle and P. Raphael.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

We will discuss special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u_0in H^{3/4^+}(R) whose restriction belongs to H^k((b,infty)) for some kinZ^+ and bin R we prove that the restriction of the corresponding solution u(cdot,t) belongs to H^k((beta,infty)) for any beta in R and any tin (0,T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schr\"odinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schr\"odinger equation, this code is compared to exact transparent boundary conditions and perfectly matched layers approaches. In addition it is shown that the Peregrine breather being discussed as a model for rogue waves can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

The purpose of this talk is to convey the idea that bifurcation theory provides a powerful tool to prove existence and orbital stability of solitons for the nonlinear Schrödinger equation. It is especially useful to obtain results for space-dependent problems, and beyond power-law nonlinearities. This will be illustrated in the case of the asymptotically linear NLS.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

The aim of this talk is to exhibit specific properties of Boussinesq type models. After recalling the usual asymptotic method leading to BT models, we will present a new asymptotic model and present a local Cauchy theory. We then provide an effective method to compute solitary waves for Boussinesq type models. We will conclude by discussing shoaling properties of such models. This is a joint work with S. Bellec.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

The Kadomtsev-Petviasvhili II equation describes wave propagating in one direction with weak transverse effect. I will explain the proof of global existence and scattering for three space dimensions. The key estimates are bilinear L^2 estimates and a delicate choice of norms. This is joint work with Junfeng Li.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

We consider a system of two coupled nonlinear Schrödinger equations in one dimension. We show the existence of solutions behaving at large time as a couple of scalar solitary waves. The proof relies on a method introduced by Martel and Merle for multi solitary waves for the scalar Schrödinger equation. Finally, we present some numerical simulations to understand more the qualitative behavior of the solitary waves.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

Many things are known about the propagation of waves at the interface of two fluids of different densities, for which dispersion plays an important role (it plays a stabilizing role controlling Kelvin-Helmholtz instabilities and balances the long time effects of the nonlinearities). When a flow is continuously stratified, the notion of wave is less clear, as well as the nature of dispersive effects. We show that they are encoded in a Sturm Liouville problem and are therefore of 'nonlocal type'; we also derive simpler, local, asymptotic models. This is a joint work with JC Saut and B. Desjardins.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

Kelvin-Helmholtz instabilities arise when a sufficiently strong shear velocity lies at the interface between two layers of immiscible fluids. The typical wavelength of the unstable modes are very small, which goes against the natural shallow-water assumption in oceanography. As a matter of fact, the usual shallow-water asymptotic models fail to correctly reproduce the formation of KH instabilities. With this in mind, our aim is to motivate and study a new class of shallow-water models with improved dispersion behavior. This is a joint work with Samer Israwi and Raafat Talhouk.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

We explain how we can get a large time of existence for the Water-Waves equation with large topography variations. We explain the method on the simplier example of the Shallow-Water equation and then present its implementation for the WW equations itselves.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

I will discuss some existence results for solitary waves with surface tension on a three-dimensional layer of water of finite depth. The waves are fully localized in the sense that they converge to the undisturbed state of the water in every horizontal direction. The existence proofs are of variational nature and different methods are used depending on whether the surface tension is weak or strong. In the case of strong surface tension, the existence proof also gives some information about the stability of the waves. The solutions are to leading order described by the Kadomtsev-Petviashvili I equation (for strong surface tension) or the Davey-Stewartson equation (for weak surface tension). These model equations play an important role in the theory. This is joint work with B. Buffoni, M. Groves and S.-M. Sun.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

"In this talk, we will present some results of inviscid limit of the 2D Navier-stokes system with data in spaces with BMO flavor. The issue of uniform (in viscosity) estimates for these equations will be also considered. It is a joint work with F. Bernicot and T. Elgindi."

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

We consider a class of pseudodifferential evolution equations of the form \(u_t +(n(u)+Lu)_x = 0\), in which L is a linear, generically smoothing, non-local operator and n is a nonlinear, local, term. This class includes the Whitham equation, the linear terms of which match the dispersion relation for gravity water waves on finite depth. In this talk we present recent results for this equation and its generalisations, including periodic bifurcation results, existence of solitary waves via minimisation, and well-posedness (local). In particular, although for small waves, small times and small frequencies this equation bears many similarities with the Korteweg—de Vries equation, it displays some very interesting differences for ’large' solutions.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

We study travelling wave solutions of a Korteweg–de Vries–Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV–Burgers equation are not in general monotone, as is the case for the corresponding classical KdV–Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV–Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

In this talk it is discussed how a non-linear system of PDEs, the Schrödinger-Klein-Gordon with Yukawa coupling, emerges naturally as the limiting dynamics of a quantum system of non-relativistic bosons coupled with a bosonic scalar field. The correspondence of the "quantum" (linear) and "classical" (nonlinear) dynamics, often assumed in physics as an heuristic theorem, is made rigorous. After a brief introduction of the quantum system (on a suitable symmetric Fock space), we identify the classical counterparts of the important objects of the quantum theory: time-evolved observables and states. In the classical context, the S-KG dynamics plays a fundamental role, and the study of its properties might provide a valuable indication of important underlying properties of the quantum system, that are much more difficult to investigate. This is a joint work with Zied Ammari.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)

We discuss a linear Fokker-Planck (FP) equation with an additional perturbation, given by a convolution with a massless kernel. In this talk we will prove the existence of a unique normalized stationary solution of the perturbed equation, and show that any solution converges towards the stationary solution with an exponential rate independent of the perturbation. The first step of the analysis consists of characterizing the spectrum of the (unperturbed) FP-operator in exponentially weighted $L^2$-spaces. In particular the FP-operator has a one-dimensional kernel (spanned by the stationary solution), possesses a spectral gap, and solutions of the unperturbed equation converge exponentially to the stationary solution. Then we demonstrate that adding a convolution with a massless kernel to the FP-operator leaves the spectrum (and the spectral gap) unchanged, i.e. the perturbed FP operator is an isospectral deformation of the FP-operator. Finally we are able to give a similarity transformation between the unperturbed and the perturbed FP operator, which proves that the corresponding semigroups have the same decay properties.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Dispersive equations with nonlocal dispersion - III" (2014)