Analysis of the Asymptotic and Analytic Behavior of Solutions to the Schrödinger-Korteweg-de Vries System

• Grant number: 24/20513-7
• Support Opportunities: Regular Research Grants
• Start date: June 01, 2025
• End date: May 31, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Argenis Jose Mendez Garcia
• Grantee: Argenis Jose Mendez Garcia
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated researchers: Marcelo Aparecido Cabral Nogueira ; Oscar Guillermo Riaño Castañeda

Abstract

This research aims to investigate the asymptotic behavior and decomposition of the solutions of the Schrödinger-Korteweg-de Vries system. The main focus is understanding how the scattering phenomenon occurs and how the solutions evolve to identify the initial conditions that favor scattering and the approximation to free states at long times. In addition, the study seeks to explore the conditions that guarantee the analyticity of the solutions, both in spatial and temporal variables, especially in scenarios with low regularity.Among the main challenges is adapting techniques already consolidated in isolated equations to the coupled system, where nonlinear interactions make the analysis more complex. Another difficulty is to ensure simultaneous analyticity in time and space variables, especially with less regular initial data. To address these issues, we will use perturbative methods and advanced harmonic analysis techniques, as well as adapt localization techniques to accurately describe the regularity of the solutions. (AU)