On Linearizing Systems of Diffusion Equations

We consider systems of diffusion equations that have considerable interest in Soil Science and Mathematical Biology and focus upon the problem of finding those forms of this class that can be linearized. In particular we use the equivalence transformations of the second generation potential system to derive forms of this system that can be linearized. In turn, these transformations lead to nonlocal mappings that linearize the original system.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Transformation methods in the study of nonlinear partial differential equations

Transformation methods are perhaps the most powerful analytic tool currently available in the study of nonlinear partial differential equations. Transformations may be classified into two categories: category I includes transformations of the dependent and independent variables of a given partial differential equation and category II additionally includes transformations of the derivatives of the dependent variables. In part I of this thesis our principal attention is focused on transformations of the category I, namely point transformations. We mainly deal with groups of transformations. These groups enable us to derive similarity transformations which reduce the number of independent variables of a certain partial differential equation. Firstly, we introduce the concept of transformation groups and in the analysis which follows three methods for determining transformation groups are presented and consequently the corresponding similarity transformations are derived. We also present a direct method for determining similarity transformations. Finally, we classify all point transformations for a particular class of equations, namely the generalised Burgers equation. Bäcklund transformations belong to category II and they are investigated in part II. The first chapter is an introduction to the theory of Bäcklund transformations. Here two different classes of Bäcklund transformations are defined and appropriate example are given. These two classes are considered in the proceeding analysis, where we search for Bäcklund transformations for specific classes of partial differential equations

The Toda lattice is super-integrable

We prove that the classical, non-periodic Toda lattice is super-integrable. In other words, we show that it possesses 2N-1 independent constants of motion, where N is the number of degrees of freedom. The main ingredient of the proof is the use of some special action--angle coordinates introduced by Moser to solve the equations of motion.Comment: 8 page

Non-Lie Reduction Operators and Potential Transformations for a Special System with Applications in Plasma Physics

Non-Lie reduction operators, also known as nonclassical symmetries, are derived for special systems that appear in Plasma Physics. These operators are used to construct similarity mappings, which reduce the systems under study into systems of ordinary differential equations. Furthermore, potential equivalence transformations are presented. Based on these results, a number of exact solutions are constructed

Transformation Properties of a Class of Variable Coefficient Boiti–Leon–Manna–Pempinelli Equations

We derive the enhanced Lie group classification for a general class of variable coefficient Boiti–Leon–Manna–Pempinelli equations. This task is achieved with the use of the equivalence group admitted by the class. Using the admitted equivalence group, we transform the general class into a much simpler class of equations. Additionally, examples of non-Lie reduction operators are presented