Painlevé analysis and transformations for nonlinear partial differential equations

Sammanfattning: Nonlinear partial differential equations play a fundamental role in the description of many physical models. In order to get a complete understanding of the phenomena which are modeled it is important to obtain exact analytic solutions. In this thesis several methods are investigated to construct analytic solutions and to classify nonlinear partial differential equations with respect to those methods. The main emphasis is on the Painlevé analysis and transformations of partial differential equations (PDEs). The Painlevé analysis for PDEs was introduced by a group of American Mathematicians in 1983. Since then many integrable equations are studied by use of this analysis. For example this analysis plays a major role in the study of soliton equations and other integrable PDEs. More recently several scientists have become interested in extending the analysis to non-integrable equations. In this thesis we investigate several non-integrable PDEs of Mathematical Physics such as the Boltzmann equations and d'Alembert-type wave equations in multidimensions. These equations model different physical phenomena in particle dynamics and wave motion. We show that the Bateman equation provides a structure for the singularity manifold for some classes of equations. This fact is applied to obtain several new solutions to the equations of interest. The second major theme of this thesis is the transformation properties of nonlinear PDEs. Two-dimensional d'Alembert-type equations invariant under Lie symmetry subalgebras of Lorenz- and conformal-type are classified. A hierarchy of linearizable evolution equations is also derived by use of a generalized hodograph transformation. This result sets the stage for a new classification of linearizable evolution equations.