The nonlinear shallow water equations and the Green-Naghdi equations are the most commonly used models to describe coastal flows. A natural question is therefore to investigate their behavior at the shoreline, i.e. when the water depth vanishes. For the nonlinear shallow water equations, this problem is closely related to the vacuum problem for compressible Euler equations, recently solved by Jang-Masmoudi and Coutand-Shkoller. For the Green-Naghdi equation, the analysis is of a different nature due to the presence of linear and nonlinear dispersive terms. We will show in this talk how to address this problem.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

Using constrained minimisation and a decomposition in Fourier space, we prove that the Kadomtsev-Petviashvili (KPI) equation modified with the exact dispersion relation from the gravity-capillary water-wave problem admits a family of small solitary solutions, approximating these of the standard KPI equation. The KPI equation, as well as its fully dispersive counterpart, describes gravity-capillary waves with strong surface tension. This is joint work with Mark Groves, Saarbrücken

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We will motivate and study a model for the propagation of surface gravity waves, which can be viewed as a fully nonlinear bi-directional Whitham equation. This model belongs to a family of systems of Green-Naghdi type with modified frequency dispersion. We will discuss the well-posedness of such systems, as well as the existence of solitary waves. The talk will be based on a work in collaboration with Samer Israwi and Raafat Talhouk (Beirut) and another in collaboration with Dag Nilsson and Erik Wahlén (Lund)

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We consider the classical gravity-capillary water-wave problem in its usual formulation as a three-dimensional free-boundary problem for the Euler equations for a perfect fluid. A solitary wave is a solution representing a wave which moves in a fixed direction with constant speed and without change of shape; it is fully localised if its profile decays to the undisturbed state of the water in every horizontal direction. The existence of fully localised solitary waves has been predicted on the basis of simpler model equations, namely the Kadomtsev-Petviashvili (KP) equation in the case of strong surface tension and the Davey-Stewartson (DS) system in the case of weak surface tension. In this talk we confirm the existence of such waves as solutions to the full water-wave problem and give rigorous justification for the use of the model equations.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

In this talk we will review some long time existence results for the abcd-Boussinesq systems. We will discuss both the Sobolev and the nonlocalized, bore-type initial data cases. The main idea in order to get a priori estimates is to symmetrize the family of systems of equations verified by the frequencies of magnitude 2^{j} of the unknowns for each j¡Ý0. For the bore-type case, an additional decomposition of the initial data into low-high frequencies is needed in order to tackle the infinite-energy aspect of these kind of data.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We justify rigorously an Isobe-Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an error of order $delta^2$, where $delta$ is a small nondimensional parameter defined as the ratio of the typical wavelength to the mean depth. The Green-Naghdi equations are known as higher order approximate equations to the water wave equations with an error of order $delta^4$. In this talk I report that the Isobe-Kakinuma model is a much higher approximation to the water wave equations with an error of order $delta^6$.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We will discuss modified scattering properties, for small Solutions and/or in the vicinity of a solitary waves for model dispersive equations in dimension one. We will mainly focus on the modified Korteweg de Vries equation and the cubic Nonlinear Schrodinger equation with potential. Joint works with P. Germain and F. Pusateri.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr ̈odinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well- posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension d ¡Ý 3 for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if d ¡Ý 5, and a careful study of the nonlinear structure of the quadratic terms in dimension 3 and 4 involving the theory of space time resonance.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

Large amplitude internal waves in a three-layer flow confined between two rigid walls will be examined in this talk. The mathematical model under consideration arises as a particular case of the multi-layer model proposed by Choi (2000) and is an extension of the two-layer MCC (Miyata-Choi-Camassa) model. The model can be derived without imposing any smallness assumption on the wave amplitudes and is well-suited to describe internal waves within a strongly nonlinear regime. We will investigate its solitary-wave solutions and unveil some of their properties by carrying out a critical point analysis of the underlying dynamical system.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

We establish the existence of solitary waves for two classes of two-layers systems modeling the propagation of internal waves. More precisely we consider the Boussinesq-Full dispersion system and the Intermediate Long Wave (ILW) system together with its Benjamin-Ono (B0) limit. This is work in progress with Jaime Angulo Pava (USP)

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on "Recent progress on surface and internal waves models" (2017)

I will discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

I shall present some recent results obtained with F. de la Hoz about the selfsimilar solutions of the Binormal Flow, also known as the Vortex Filament Equation. Some connections with the transfer of energy in the case when the filament is a regular polygon will be also made.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

By combining the Kenig-Merle approach with a suitable inequality proved by Tao we deduce that solutions to gKdV, in the L^2-supercitical regime, scatter to free waves for large times.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We discuss some essential features of solitons for the energy-critical half-wave maps equation. Furthermore, we will present a Lax pair structure and explain its applications to understanding the dynamics. The talk is based on joint work with P. Gérard (Orsay) and A. Schikorra (Pittsburgh).

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

In this talk we will discuss some recent improvements on well-known decay estimates for nonlinear dispersive and wave equations in 1D with supercritical decay, or no decay at all. Using Virial estimates, we will get local decay where standard dispersive techniques are not available yet. These are joint works with M.-A. Alejo, M. Kowalczyk, Y. Martel, F. Poblete, and J.-C. Pozo.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We will see various notion of nondispersive solution in the case of the energy criticl wave equation and applications.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We consider the $L^2$ critical gKdV equation with a saturated perturbation. In this case, all $H^1$ solution are global in time. Our goal is to classify the asymptotic dynamics for solutions with initial data near the ground state. Together with a suitable decay assumption, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave, whose $H^1$ norm is of size $gamma^{-2/(q-1)}$, as $gammarightarrow0$; (ii) the solution is always in a small neighborhood of the modulated family of solitary waves, but blows down at $+infty$; (iii) the solution leaves any small neighborhood of the modulated family of the solitary waves. This extends the result of classification of the rigidity dynamics near the ground state for the unperturbed $L^2$ critical gKdV (corresponding to $gamma=0$) by Martel, Merle and Rapha"el. It also provides a way to consider the continuation properties after blow-up time for $L^2$ -crtitical gKdV equations.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We consider two non-variational semilinear parabolic systems, with different diffusion constants between the two components. The reaction terms are of power type in the first system. They are of exponential type in the second. Using a formal approach, we derive blow-up profiles for those systems. Then, linearizing around those profiles, we give the rigorous proof, which relies on the two-step classical method: (i) the reduction of the problem to a finite-dimensional one, then, (ii) the proof of the latter thanks to Brouwer's lemma. In comparison with the standard semilinear heat equation, several technical problems arise here, and new ideas are needed to overcome them. This is a joint work with T. Ghoul and V.T. Nguyen from NYU Abu Dhabi.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

This talk is about singularity formation for solutions to $$ (*) partial_{t}u+upa_x u-pa_{yy}u=0, (x,y) in mathbb R^2 $$ which is a simplified model of Prandtl's boundary layer equation. Note that it reduces to Burgers equation for $y$-independent solutions $u(t,x,y)=v(t,x)$. We will first recast the well-known shock formation theory for Burgers equation using the framework of self-similar blow-up. This will provide us with an analytic framework to study the effect of the transversal viscosity. The main result (still work in progress) is the construction and precise description of singular solutions to $(*)$. This is joint work with T.E. Ghoul and N. Masmoudi.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We solve the cubic nonlinear Schrödinger equation on $mathbb R$ with initial data a sum of Diracs. Then we describe some consequences for a class of singular solutions of the binormal flow, that is used as a model for the vortex filaments dynamics in 3-D fluids and superfluids. This is a joint work with Luis Vega.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

We consider the linear wave equation outside a compact, strictly convex obstacle in R^3 with smooth boundary and we show that the linear wave flow satisfies the dispersive estimates as in R^3 (which is not necessarily the case in higher dimensions).

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)
• Event: Workshop on „Propagation of Singularities in Dispersive PDEs“ (2017)

In this talk, I will show our recent work on 3D magnetic nanostructures for applications in spintronics. We are developing 3D nano-printing methods based on focused electron beams [2]. In particular, we have achieved great control over the growth of 3D magnetic nanowires for domain wall studies [3]. Advanced magnetic microscopy experiments reveal the magnetic state and magnetisation reversal mechanism of the wires, dominated by their geometry and metallic composition [4]. Recent results also show how controllable domain wall motion along the whole space becomes now possible [5]. This has been realised by development of new methods for 3D nano-printing and magneto-optical detection of 3D nanostructures. During the talk, I will discuss novel methodologies to characterise 3D nanomagnets, including magneto-optical, electron and X-ray microscopy. I will also highlight key challenges and opportunities of 3D nanomagnetism.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)

In this talk, I will present an overview/review of progress in articial gauge fields in quantum systems. I will start with the underlying first principles with the seminal paper of Berry, the Berry or Geometric phase. Following a few month after its publication Wilczek and Zee concluded with Berry's results, that non-Abelian gauges fields can naturally emerge from the adiabatic development of simple quantum systems. I will mainly focus on how ultracold atomic systems can be prepared such that a mapping to a ultracold atoms behaving like charged particles in a magnetic field. The induced gauge field whether abelian or non-Abelian introduces a space dependent coupling between the dressed states of the ultracold atoms. This provides motivation for extending MCTDH-X to tackle quantum systems with artificial gauge fields where the spatial dynamics of the dressed states or pseudo-spins can be studied in great detail. This could open up interesting physics that could potentially be observed in the experiment.

• Thematic program: PDEs in quantum physics, materials and micromagnetism (2017/2018)