The mathematical description of physical and biological phenomena is often performed through mathematical modeling involving differential equations. The more realistic a model becomes, the more likely it is to incorporate non-linear effects. An example of nonlinearity is the growth of a given population of individuals in a medium: at low time scales, the population grows at a rate proportional to the number of individuals. However, after a specific time, the number of individuals has an opposite effect on their growth, thus combining a linear effect (growth) with a non-linear one (control).

Nonlinear phenomena are quite common in science and a vibrant field of research in pure and applied mathematics. It should be emphasized that the mathematical study of non-linear phenomena is not a simple task, often requiring the use of special analytical techniques or numerical and computational methods.

This project aims to continue several lines of research that has been developing in the past ten years in the area of non-linear phenomena. It focuses on the following topics:

1) Mathematical aspects of integrable systems, i.e., systems that have infinite symmetries, conservation laws, recursion operators and bi-Hamiltonian formulation. Of particular interest is the investigation of: a) non-evolutionary equations or systems of the Camassa-Holm type; b) algebral-geometric properties of new evolutionary equations; c) mechanical-geometric properties of Ibragimov’s conservation law theory;

2) study and modelling of biological phenomena, with special emphasis on the modelling of cancer, using numerical and computational methods for simulations, and Lie symmetries to obtain qualitative information and explicit analytical solutions;

3) study of elliptical systems on manifolds, where Lie symmetries would be the basic tool to determine the group of invariance and construction of conservation laws that would enable a qualitative analysis of the properties of the proposed solutions for the investigated problems.

**Coordinator:** Prof. Dr. Igor Leite Freire

**E-mail:** igor.freire@ufabc.edu.br

**Coordinator’s Curriculum Lattes (research projects, publications and academic info)**

**Coordinator’s research grants, scholarships and main publications (FAPESP)**