Estudo das equações diferenciais via sistemas diferenciais exteriores e suas reduções por simetrias.

Abstract

In general terms, this dissertation aims to study, from the geometric perspective of jet spaces and exterior differential systems, the differential equations and the relationships that can be established between their solutions. As we will see, from the process of reduction by symmetries of exterior differential systems, we can systematically construct transformations that relate solutions of equations obtained by a reduction process so that, in this way, we can establish Bäcklund transformations between differential equations. The work is subdivided into five chapters. Chapter 1 presents some prerequisites on smooth manifolds, taking the opportunity to introduce the main notations and conventions used throughout the text. Succesively, in Chapter 2, some basic results are shown in the context of submersions, and, more specifically, some facts about fiber bundles are established that will be useful for the development of the following chapters. Chapter 3, taking advantage of the theoretical tools in previous chapters, introduces the jet spaces of sections of a fiber bundle. Thus, the interpretation of differential equations as submanifolds of these geometric environments is discussed, and based on this, some essential notions of symmetry are presented, together with some basic results. Chapter 4 addresses the symmetry reduction of differential equations form the point of view of the theory of exterior differential systems. Therefore, after a short introduction to the formalism of exterior differential systems, their applications to the study of differential equations are immediately discussed, aiming to elucidate above all the aspects that play a central role in symmetry reduction. Finally, Chapter 5 is devoted to the main topic of this work, which is the discussion of a useful method for obtaining Bäcklund transformations through different reductions of the same exterior differential system.