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Minimal solutions to nonlinear systems of PDEs

Project description

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.

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.

Coordinator

BCAM - BASQUE CENTER FOR APPLIED MATHEMATICS
Net EU contribution
€ 160 932,48
Address
AL MAZARREDO 14
48009 Bilbao
Spain

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Region
Noreste País Vasco Bizkaia
Activity type
Research Organisations
Links
Total cost
€ 160 932,48