Symmetries, Integrability and Exact Solutions for Nonlinear Systems

TL;DR

This paper provides an overview of integrability in nonlinear systems, emphasizing symmetry methods and illustrating their application through two models, with some novel results on optimal systems of solutions.

Contribution

It introduces new findings on optimal systems of solutions and demonstrates how symmetry methods can be effectively applied to nonlinear dynamical systems.

Findings

New results on optimal systems of solutions.

Application of symmetry methods to Ricci flow and convection-diffusion models.

Illustration of the procedure for nonlinear systems.

Abstract

The paper intends to offer a general overview on what the concept of integrability means for a nonlinear dynamical system and how the symmetry method can be applied for approaching it. After a general part where key problems as direct and indirect symmetry method or optimal system of solutions are tackled out, in the second part of the lecture two concrete models of nonlinear dynamical systems are effectively studied in order to illustrate how the procedure is working out. The two models are the 2D Ricci flow model coming from the general relativity and the 2D convective-diffusion equation . Part of the results, especially concerning the optimal systems of solutions, are new ones. Keywords: Lie symmetries, invariants, similarity reduction.