Simple Non Linear Klein-Gordon Equations in 2 space dimensions, with long range scattering

We establish that solutions, to the most simple NLKG equations in 2 space dimensions with mass resonance, exhibits long range scattering phenomena. Modified wave operators and solutions are constructed for these equations. We also show that the modified wave operators can be chosen such that they linearize the non-linear representation of the Poincar\'e group defined by the NLKG.Comment: 19 pages, LaTeX, To appear in Lett. Math. Phy

Scattering and small data completeness for the critical nonlinear Schroediger equation

We prove Asymptotic Completeness of one dimensional NLS with long range nonlinearities. We also prove existence and expansion of asymptotic solutions with large data at infinity

Interaction of vortices in viscous planar flows

We consider the inviscid limit for the two-dimensional incompressible Navier-Stokes equation in the particular case where the initial flow is a finite collection of point vortices. We suppose that the initial positions and the circulations of the vortices do not depend on the viscosity parameter \nu, and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex system is well-posed on the interval [0,T]. Under these assumptions, we prove that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a superposition of Lamb-Oseen vortices whose centers evolve according to a viscous regularization of the point vortex system. Convergence holds uniformly in time, in a strong topology which allows to give an accurate description of the asymptotic profile of each individual vortex. In particular, we compute to leading order the deformations of the vortices due to mutual interactions. This allows to estimate the self-interactions, which play an important role in the convergence proof.Comment: 39 pages, 1 figur

Resonance-free Region in scattering by a strictly convex obstacle

We prove the existence of a resonance free region in scattering by a strictly convex obstacle with the Robin boundary condition. More precisely, we show that the scattering resonances lie below a cubic curve which is the same as in the case of the Neumann boundary condition. This generalizes earlier results on cubic poles free regions obtained for the Dirichlet boundary condition.Comment: 29 pages, 2 figure

Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations

In this paper, we consider Hartree-type equations on the two-dimensional torus and on the plane. We prove polynomial bounds on the growth of high Sobolev norms of solutions to these equations. The proofs of our results are based on the adaptation to two dimensions of the techniques we previously used to study analogous problems on $S^1$, and on $\mathbb{R}$.Comment: 38 page

Normal Forms for Semilinear Quantum Harmonic Oscillators

We consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\geq 2$. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions

Twitter permeability to financial events: an experiment towards a model for sensing irregularities

There is a general consensus of the good sensing and novelty character- istics of Twitter as an information media for the complex fi nancial market. This paper investigates the permeability of Twitter sphere, the total universe of Twitter users and their habits, towards relevant events in the financial market. Analysis shows that a general purpose social media is permeable to fi nancial-specifi c events and establishes Twitter as a relevant feeder for taking decisions regarding the fi nancial market and event fraudulent activities in that market. However, the provenance of contributions, their diferent levels of credibility and quality and even the purpose or intention behind them should to be considered and carefully contemplated if Twitter is used as a single source for decision taking. With the overall aim of this research, to deploy an architecture for real-time monitoring of irregularities in the financial market, this paper conducts a series of experiments on the level of permeability and the permeable features of Twitter in the event of one of these irregularities. To be precise, Twitter data is collected concerning an event comprising of a specifi c financial action on the 27th January 2017: the announcement about the merge of two companies Tesco PLC and Booker Group PLC, listed in the main market of the London Stock Exchange (LSE), to create the UK's Leading Food Business. The experiment attempts to answer two research questions which aim to characterize the features of Twitter permeability to the fi nancial market. The experimental results con rm that a far-impacting financial event, such as the merger considered, caused apparent disturbances in all the features considered, that is, information volume, content and sentiment as well as geographical provenance. Analysis shows that despite, Twitter not being a specifi c fi nancial forum, it is permeable to financial events