In this talk, we investigate the properties of solitonic structures arising in quadratic media. More precisely, we look for stationary states in the context of normal or anomalous dispersion regimes, that lead us to either elliptic or nonelliptic systems and we address the problem of orbital stability. Finally, we present some numerical experiments in order to compute localized states for several regimes.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We will survey recent results and open problems on various nonlocal "full dispersion" models of surface water waves.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

The usual model of nonlinear optics given by the cubic NLS equation is too crude to describe large intensity phenomenas such as filamentation, which modifies the focusing of laser beams. I shall explain how to derive some more appropriate variants of the NLS model from Maxwell's equations, using improved approximations of the original dispersion relation or taking ionization effects into account. I shall provide rigorous error estimates for the models considered, and also discuss some open problems related to these modified NLS equations. This is joint work with David Lannes and Jeremie Szeftel.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

Terahertz (THz) emissions have nowadays important applications such as security screening and imaging. Laser-gas interaction reveals itself to be a promising technique to generate broadband and intense THz sources suitable for these applications. In this talk, I will explain recent mathematical models and their underlying physics explaining the THz radiation generated when ultrafast laser pulses ionize a gas at high intensities. Solutions to the model equations will be compared with direct numerical simulations.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation with a critical nonlinearity and prove the asymptotic stability of solutions in the neighborhood of the self similar solutions under the non zero mass condition and the smallness on the data.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin as a model of laser-plasma interaction. In this talk, we prove the well-posedness of this system for the periodic initial data. In particular, if the coefficients of Laplacian satisfy some conditions, then the well-posedness is proved at the scaling critical regularity by using U^2 and V^2 spaces.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider the 1D Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we study existence and stability of the black solitons with a combination of variational and perturbation arguments. This is a joint work with Isabella Ianni and Julien Royer.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

In this talk, I will explain the Proudman resonance. It is a resonant respond in shallow waters of a water body on a traveling atmospheric disturbance when the speed of the disturbance is close to the typical water wave velocity. In order to explain this phenomenon, I will prove a local well-posedness of the water waves equations with a non constant pressure at the surface, taking into account the dependence of small physical parameters. Then, I will justify mathematically the historical work of Proudman. Finally, I will study the linear water waves equations and I will give dispersion estimates in order to extend The Proudman resonance to deeper waters. To complete these asymptotic models, I will show some numerical simulations.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We consider a system of nonlinear Schrodinger equations with cubic nonlinearity, called a coherently coupled NLS system (CCNLS) in nonlinear optics, in one space dimension. We study orbital stability and instability of standing wave solutions of (CCNLS), and prove similar results to Colin and Ohta (2012) which studies a system of NLS equations with quadratic nonlinearity. This is a joint work with Shotaro Kawahara (Tokyo University of Science).

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

The equation in (BI) appears for instance in the Born-Infeld nonlinear electromagnetic theory: in the electrostatic case it corresponds to the Gauss law in the classical Maxwell theory and so  is the electric potential and  is an assigned extended charge density. We discuss existence, uniqueness and regularity of the solution of (BI). The results have been obtained in a joint work with Denis Bonheure and Pietro d’Avenia.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

A new NLS type equation is employed for modeling long-range interactions in nonlinear optics, in a collaboration with experimental physicists. It is of quasilinear type and models fluctuations around a 'continuous-wave polariton' which are chosen according to Bogoliubov theory. We present a numerical discretization method and simulation results. Mathematical theory for this equation is work in progress.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

In this talk, we consider a quasiinear Schrodinger equation which appears in the study of plasma physics. We are interested in the uniqueness of ground states without assuming any restriction on a physical parameter. We also study asymptotic behavior of ground states as the parameter goes to zero.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)

We show that space-time resonances induce high-frequency instabilities in the two-fluid Euler-Maxwell system. This implies in particular that the Zakharov approximation to Euler-Maxwell is stable if and only if the group velocity vanishes. The instability proof relies on a short-time representation formula for the flows of pseudo-differential operators of order zero. This is joint work with Eric Dumas (Grenoble) and Lu Yong (Prague).

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2015/2016)
• Event: Workshop on "Quasilinear and nonlocal nonlinear Schrödinger equations" (2015)