Grupo de Ecuaciones Diferenciales y Aplicaciones (GEDA)

Many phenomena of real life, experimental sciences and technology are modelled using partial differential equations (PDEs): as examples all those involving atoms and molecules, fluids, weather, financial markets, populations, …

In GEDA we deal with the mathematical study of some priority scientific questions related to different aspects of physics and ecology. Under this premise, we present different proposals with differentiated starting points, but with common threads. On the one hand, its applicability, and its clearly innovative character, putting mathematics together with other sciences of special relevance in the present day. On the other hand, its theoretical development, combines the study of non-local or higher order operators, reaction-diffusion equations, different boundary conditions and the development of numerical models that validate the results.

Biological and Ecological models

We propose to analyse biological and ecological models both theoretically and numerically, to find an answer to some crucial questions such as: how do populations interact? What happens if they live in separate regions and interact through a boundary? Is it possible to find safe areas?

Theoretical analysis of these and related questions should provide us with a series of protocols to increase species productivity or, perhaps, to avoid unpleasant situations in which productivity is overestimated, ultimately causing species extinction. These issues
seem to be crucial nowadays given the situation we live in because of the industrial revolution, the development of medicine, transports or communications, which among other things have shaped the welfare state and the exponential growth of the population, with the ecological consequences that this entails.

From the mathematical point of view, we would like to point out that it is a novel proposal to combine in the same problem the study of the behaviour of populations
with heterogeneities (spatial or temporal heterogeneities) with nonlocal or higher-order diffusions. To date, there is no literature on this subject; however, we believe that this type of approach will allow us to design more realistic models.

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Cristina Brändle (cristina.brandle@uc3m.es)

Quantum Mechanics through the Calculus of Variations

Many physical models, such as in nonlinear optics or in multispecies Bose-Einstein condensates (BEC), consist of systems of Schrödinger-type equations coupled through nonlinear terms, or coupled nonlinear Schrödinger-Korteweg de Vries equations related with the study of Tsunamis. If one considers a particular type of solutions, the solitary wave solutions, then these systems give rise to elliptic systems with nonlinear coupling.

The story of “BEC” goes back to the 1920's when Satyendra Nath Bose (1894-1974) sent a paper to be published in a scientific journal and it was rejected. Bose revised his paper and sent it to Albert Einstein (1879-1955) in 1924. Einstein saw the potential of the investigation by Bose and extended its results with new applications. This led them to meet in Berlin 1925 and to finish the definition of the known “Bose-Einstein Condensates” model. The BEC were shown experimentally in 1990’s by the German Wolfgang Ketterle and the Americans Eric A. Cornell and Carl E. Wieman. In 2001, Ketterle-Cornell-Wieman won the Nobel Prize in Physics by the conquest of the “BEC”, the fifth state of matter which holds at ultra-cold temperature in which atoms behave like a unique super-atom following the quantum-mechanic laws. That is, in the BEC state, all atoms condensate into a so-called quantum-mechanical ground state.

Inspired by previous works, we will deal with systems of two nonlinear Schrödinger equations or systems with one of the components being nonlinear Schrödinger equation while the other is a Korteweg de Vries equation, defined in bounded domains under mixed boundary conditions.

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)

Optimization Problems

The calculus of variations is a field that deals with the optimization of certain types of functions called functionals. These functionals, related to the energy of a certain system, appear when considering, for example, minimal surfaces, transport optimization or some optical phenomena and, therefore, one type of interesting solutions are the so-called minimum energy solutions or minimizers. Two questions naturally arise: Do solutions exist in general? And if so, are the solutions of regular problems in the calculus of variations always necessarily regular?

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Eduardo Colorado (ecolorad@math.uc3m.es)

Problems assuming membrane regions

We consider systems of reaction-diffusion equations defined in different domains and coupled only through a common boundary. The examples we are dealing with model, for example, ecological or migratory issues, in which different populations live separately, in different regions of a territory, and interact only at a common border; proteins crossing a biological membrane in a cell; or problems of quantum mechanics where we find different quantum systems on different sides of the domain, while connected by the interaction through a barrier acting as insulation region. As a canonical model, we first consider the Kedem-Katchalsky conditions which were introduced in a thermodynamic context. They are represented by the equality of the flow going through a membrane and are proportional to the difference of densities on both sides of the domain. The main aim will be to to provide answers concerning the existence, multiplicity and regularity of solutions of systems involving those conditions on the interface region.

Contact: Pablo Álvarez-Caudevilla (pacaudev@math.uc3m.es), Cristina Brändle (cristina.brandle@uc3m.es)

Numerical Simulation of Biological Neuron Networks

This master thesis focuses on the numerical simulation of the nonlinear noisy leaky integrate-and-fire (NNLIF) neuron model, emphasised in the development and implementation of numerical schemes for solving the partial differential equations (PDEs) governing neuronal dynamics. The study extends to the incorporation of stochastic differential equations (SDEs) to account for the microscopic noise inherent in the neural systems. The primary objective of this work is to implement and test numerical methods that enable the efficient and accurate solution of these complex equations. The thesis also aims to provide a user-friendly graphical interface for real-time simulations, allowing users to manipulate and explore various model parameters such as synaptic strength. By leveraging this interface, users can visualize neuronal activity under different conditions and gain insight into neuron network behavior. This work is expected to contribute to the understanding of how neural circuits respond to fluctuating inputs and to offer valuable tools for future computational neuroscience research.

Contact: Alejandro Ramos Lora (alramosl@math.uc3m.es)