| In this doctoral thesis, we make use of certain theoretical methods involving Lie groups in order to construct exact analytic solutions of mathematical models related to fluid mechanics. In particular, we consider the equations of motion of the nonrelativistic Chaplygin gas and the relativistic Born-Infeld model for a scalar field, as well as certain supersymmetric generalizations. Through a comprehensive analysis of the symmetry properties and subgroup classification structure of these systems, a large number of new classes of invariant and partially invariant solutions are obtained for the classical Chaplygin and Born-Infeld theories in (1 + 1) dimensions, and for the supersymmetric extensions in (1 + 1) and (2 + 1) dimensions.; The main body of this thesis is made up of three original contributions, which consist of articles published or submitted to scientific journals.; In the first article, a procedure based on the symmetry reduction method is used to obtain invariant solutions of the (1 + 1)-dimensional Chaplygin and Born-Infeld equations. This procedure makes systematic use of the one-dimensional subalgebras of the Lie symmetry algebra of these equations in order to construct invariant analytic solutions. We obtain algebraic, rational and solitonic type solutions, including bumps, kinks and multiple waves. This article was published in the Journal of Mathematical Physics in July 2003.; In the second article, the concept of partially invariant solutions is applied to the Chaplygin and Born-Infeld models in (1 + 1) dimensions. Using a general systematic approach based on the classification of the two-dimensional subalge-bras of the symmetry Lie algebra, partially invariant solutions with defect delta = 1 are constructed. We obtain travelling waves, centered waves, solitonic type solutions including kinks and bumps, and solutions solved in terms of Jacobi elliptic functions. This article was published in the Journal of Mathematical Physics in August 2004.; In the third article, we present a procedure which allows us to determine solutions of the supersymmetric Chaplygin gas models in (1 + 1) and (2 + 1) dimensions. Through the use of a generalized Legendre transform, a number of analytic solutions of the linear supersymmetric model were found. In addition, certain basic elements of a possible extension of our method to the planar supersymmetric model are presented. The results were submitted to the Journal of Physics A in January 2005. |
