Opis projektu
Minimalistyczne rozwiązania nieliniowych równań różniczkowych cząstkowych
Projekt MinSol-PDEs finansowany ze środków działania „Maria Skłodowska-Curie” obejmie systematyczne badania minimalnych rozwiązań dla dużej klasy nieliniowych równań różniczkowych cząstkowych. Część tych badań będzie ukierunkowana na problemy przejść fazowych opisanych równaniem Allena-Cahna. Badacze chcą doprowadzić do redukcji równania do układu hamiltonowskiego w celu stworzenia nowej klasy minimalnych rozwiązań i lepszego zrozumienia warunków implikujących redukcję zmiennych. Inny obszar badań będzie skoncentrowany na równaniu Painlevé, które odgrywa kluczową rolę w wielu dziedzinach, w tym losowych macierzach, nadprzewodnictwie czy układach dynamicznych. Głównym celem jest klasyfikacja i badanie minimalnych rozwiązań układów typu Painlevé w niskich wymiarach.
Cel
The aim of this proposal is to provide a systematic study of minimal solutions for a large class of nonlinear systems of PDE. Namely we will construct minimal solutions with predefined characteristics and investigate their qualitative properties, addressing the fundamental challenges that appear in the case of systems and which cannot be tackled with tools from the scalar case.
The first part focuses on phase transition problems described by the Allen-Cahn system. This is a hot and difficult topic linking PDE with the theory of minimal surfaces. The main idea is to reduce the Allen-Cahn system to a Hamiltonian system in order to construct new classes of minimal solutions, and understand the conditions implying the reduction of variables (vector analog of the celebrated De Giorgi conjecture).
In the second part, our focus is on the Painlevé equation which plays a crucial role in areas as diverse as random matrices, integrable systems, and superconductivity. The objective is to classify and investigate the minimal solutions of Painlevé-type systems in low dimensions. These have direct applications in the study of vortices in liquid crystals and Bose-Einstein condensates. The proposed approach connects the Painlevé equation with a singular problem, easier to study.
The fellow has a strong research record on the Allen-Cahn system (a book + 6 papers), and has also worked on the Ginzburg-Landau model of liquid crystals. On the one hand, he will develop his own innovative approaches to the proposed problems, and transfer his expertise to the host. On the other hand, at BCAM and through a secondment, he will link his previous research on liquid crystals to other alternative models (for which the supervisor is a world-leading expert), and to the theory of Bose-Einstein condensates. He will also acquire new skills in simulation and computation. The achievement of this project will reinforce Fellow's reputation and support him in obtaining a strong academic position.
Dziedzina nauki
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesphysical sciencescondensed matter physicsbose-einstein condensates
- natural sciencesphysical scienceselectromagnetism and electronicssuperconductivity
- engineering and technologymaterials engineeringliquid crystals
Program(-y)
Temat(-y)
System finansowania
MSCA-IF-EF-ST - Standard EFKoordynator
48009 Bilbao
Hiszpania