Symmetries, Equivalence and Decoupling of First Order PDE's

Abstract

The present Ph.D. Thesis is concerned with first order PDE's and to the structural conditions allowing for their transformation into an equivalent, and somehow simpler, form. Most of the results are framed in the context of the classical theory of the Lie symmetries of differential equations, and on the analysis of some invariant quantities. The thesis is organized in 5 main sections. The first two Chapters present the basic elements of the Lie theory and some introductory facts about first order PDE's, with special emphasis on quasilinear ones. Chapter 3 is devoted to investigate equivalence transformations, i.e., point transformations suitable to deal with classes of differential equations involving arbitrary elements. The general framework of equivalence transformations is then applied to a class of systems of first order PDE's, consisting of a linear conservation law and four general balance laws involving some arbitrary continuously differentiable functions, in order to identify the elements of the class that can be mapped to a system of autonomous conservation laws. Chapter 4 is concerned with the transformation of nonlinear first order systems of differential equations to a simpler form. At first, the reduction to an equivalent first order autonomous and homogeneous quasilinear form is considered. A theorem providing necessary conditions is given, and the reduction to quasilinear form is performed by constructing the canonical variables associated to the Lie point symmetries admitted by the nonlinear system. Then, a general nonlinear system of first order PDE's involving the derivatives of the unknown variables in polynomial form is considered, and a theorem giving necessary and sufficient conditions in order to map it to an autonomous system polynomially homogeneous in the derivatives is established. Several classes of first order Monge-Ampere systems, either with constant coefficients or with coefficients depending on the field variables, provided that the coefficients entering their equations satisfy some constraints, are reduced to quasilinear (or linear) form. Chapter 5 faces the decoupling problem of general quasilinear first order systems. Starting from the direct decoupling problem of hyperbolic quasilinear first order systems in two independent variables and two or three dependent variables, we observe that the decoupling conditions can be written in terms of the eigenvalues and eigenvectors of the coefficient matrix. This allows to obtain a completely general result. At first, general autonomous and homogeneous quasilinear first order systems (either hyperbolic or not) are discussed, and the necessary and sufficient conditions for the decoupling in two or more subsystems proved. Then, the analysis is extended to the case of nonhomogeneous and/or nonautonomous systems. The conditions, as one expects, involve just the properties of the eigenvalues and the eigenvectors (together with the generalized eigenvectors, if needed) of the coefficient matrix; in particular, the conditions for the full decoupling of a hyperbolic system in non-interacting subsystems have a physical interpretation since require the vanishing both of the change of characteristic speeds of a subsystem across a wave of the other subsystems, and of the interaction coefficients between waves of different subsystems. Moreover, when the required decoupling conditions are satisfied, we have also the differential constraints whose integration provides the variable transformation leading to the (partially or fully) decoupled system. All the results are extended to the decoupling of nonhomogeneous and/or nonautonomous quasilinear first order systems.