Geometric and qualitative aspects of some non-local integrable models

• Grant number: 22/00163-6
• Support Opportunities: Scholarships abroad - Research
• Start date: November 14, 2022
• End date: November 13, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Igor Leite Freire
• Grantee: Igor Leite Freire
• Host Investigator: Alexander Veselov
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Institution abroad: Loughborough University, England

Abstract

We want to continue the investigation of structural and qualitative properties of certain equations discovered in 2009 by V. Novikov, which was started through FAPESP grant 2020/02055-0. From the point of view of structural properties, our main goal is the better understanding of geometric aspects of these equations. We are particularly interested in determining which of these equations describe pseudo-spherical surfaces and are geometrically integrable. Next, we aim to study their corresponding first and second fundamental forms for the ones describing pseudo-spherical surfaces. From the view point of qualitative analysis, we are interested in determine conditions for local and global well posedness of solutions, as well as understanding the mechanisms of their blow up. We are also looking for persistence properties of the solutions, as well as finding explicit soliton solutions. These two aspects are them merged in the study of how we can understand the pseudo-spherical surfaces coming from the solutions of the equations. (AU)