| Hour | Tuesday | |||
| 9:00-9:45 | Rafael Granero | |||
| Coffee Break | ||||
| 10:15-11:00 | Raimund Bürger | |||
| 11:05-13:00 | Conexión – 2025 Ramanujan Prize Ceremony | |||
| Lunch | ||||
| 14:30-15:15 | André de Laire | |||
| Coffee Break | ||||
| 15:45-16:30 | Jessica Trespalacios | |||
| 16:45-17:30 | Sebastián Tapia | |||
9:00 – Rafael Granero-Belinchón (U. Cantabria)
On some PDE modelling nonlinear waves
In this talk we will present some new PDEs modelling nonlinear waves. These models take the form of nonlinear and nonlocal differential equations. Furthermore, we will also present some mathematical results for these equations.
9:45 – Café
10:15 – Raimund Bürger (U. de Concepción)
A multilayer shallow water model for tsunamis and coastal forest
interaction
Models and numerical methods of the impact of tsunamis on coastal forests are of vital importance for exploring the potential of coastal vegetation as a means of mitigation. Such a model is formulated as a multilayer shallow water system based on a free-surface formulation of the Euler equations for an ideal fluid. Specifically, the Euler equations are approximated by a layer averaged non-hydrostatic (LDNH) approach involving linear pressures and piecewise constant velocities. Furthermore, based on [Iimura and Tanaka, Ocean Eng. 54 (2012) 223–232] drag forces, inertia forces, and porosity are added to model the interaction with the forest. These ingredients are specified in a layer-wise manner. Thus, the vertical features of the forest are described with higher accuracy than within a single-layer approach. Projection methods for the non-hydrostatic pressure in conjunction with polynomial viscosity matrix finite volume methods [Castro and Fernández-Nieto, SIAM J. Sci. Comput. 34 (2012) A2173–A2196] are employed for the numerical solution of the multilayer model, that is for the propagation of tsunamis and coastal flooding. Experimental observations and field data are used to validate the model. In general good agreement is obtained. Work in progress addresses the extension of the model and its numerical approximation to two horizontal space dimensions.
This presentation is based on joint research with Enrique Fernández-Nieto (Universidad de Sevilla, Spain) and Jorge Moya (Universidad de Concepción).
11:05 – Conexión – 2025 Ramanujan Prize Ceremony –
13:00 – Almuerzo
14:30 André de Laire (U. Lille and INRIA Lille)
Minimizing travelling waves for the Gross-Pitaevskii equation on the 2D strip
We investigate the two-dimensional defocusing nonlinear cubic Schrödinger (Gross–Pitaevskii) equation with nonzero conditions at infinity, on a strip, i.e. on an infinite channel of finite transverse width. We establish the existence of traveling waves that minimize the Ginzburg–Landau energy at fixed momentum. We establish a sharp bifurcation from planar to multidimensional behavior. Precisely, we show that there exists a threshold value for the transverse width below which minimizers are the one-dimensional dark solitons (planar solitons), and above which they are genuinely two-dimensional dark solitons.
This is a joint work with Didier Smets and Philippe Gravejat.
15:15 – Café
15:45 – Jessica Trespalacios (U. Austral de Chile)
Nonlinear Stability of nonsingular Solitons of the Principal Chiral Field equation
We consider the Principal Chiral Field (PCF) model posed in 1+1 dimensions into the Lie group $\text{SL}(2,\mathbb R)$. In this talk we show the nonlinear stability of small enough nonsingular solitons. In particular, we are interested in the notion of orbital and asymptotic stability of special soliton solutions of PCF model with small initial data perturbations. The method of proof involves the use of vector field methods as in a previous work by Muñoz et. al. dealing with Einstein’s field equations under the Belinski-Zakharov formalism, extending for all times the size of suitable null weighted norms of the perturbations at time zero.
This is a joint work with Miguel Ángel Alejo and Claudio Muñoz.
16:45 – Sebastián Tapia-Mandiola (U. Lille and INRIA Lille)
Numerical simulations of a quasilinear Gross-Pitaevskii equation with vanishing and nonvanishing conditions at infinity
In this talk, we focus on the quasilinear Schrödinger equation with zero and nonzero conditions at infinity. This quasilinear model corresponds to a weakly nonlocal approximation of the nonlocal Gross–Pitaevskii equation, and can also be derived by considering the effects of surface tension in superfluids. We can distinguish two regimes: focusing and defocusing. In the focusing case, the existence and stability of bright solitons have already been well studied. In the defocusing case with non-zero background, a complete classification of finite-energy traveling waves has recently been achieved, leading to the existence of dark solitons. Our goal is to develop an energy-preserving numerical scheme that accurately computes approximations of the equation’s evolution. We are interested in: stability of bright solitons under collisions varying κ, short-term blow-up in the defocusing case with zero background, and long-term stability of dark solitons.