Project description DEENESFRITPL Minimal solutions for solving nonlinear partial differential equations Funded by the Marie Skłodowska-Curie Actions programme, the MinSol-PDEs project will conduct a systematic study of minimal solutions for a large class of nonlinear partial differential equations. Part of the research will be geared towards phase transition problems described by the Allen–Cahn equation. The main idea is to reduce the equation to a Hamiltonian system to construct new classes of minimal solutions and understand the conditions implying variable reduction. Another part of the research will focus on the Painlevé equation, which plays a crucial role in areas as diverse as random matrices, integrable systems and superconductivity. The main goal is to classify and investigate the minimal solutions of Painlevé-type systems in low dimensions. Show the project objective Hide the project objective Objective 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. Fields of science natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equationsnatural sciencesphysical sciencescondensed matter physicsbose-einstein condensatesnatural sciencesphysical scienceselectromagnetism and electronicssuperconductivityengineering and technologymaterials engineeringliquid crystals Programme(s) H2020-EU.1.3. - EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions Main Programme H2020-EU.1.3.2. - Nurturing excellence by means of cross-border and cross-sector mobility Topic(s) MSCA-IF-2018 - Individual Fellowships Call for proposal H2020-MSCA-IF-2018 See other projects for this call Funding Scheme MSCA-IF-EF-ST - Standard EF Coordinator BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS Net EU contribution € 160 932,48 Address AL MAZARREDO 14 48009 Bilbao Spain See on map Region Noreste País Vasco Bizkaia Activity type Research Organisations Links Contact the organisation Opens in new window Website Opens in new window Participation in EU R&I programmes Opens in new window HORIZON collaboration network Opens in new window Total cost € 160 932,48