Connections Between Symmetries and Conservation Laws

This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for any given system of differential equations. This method yields the multipliers for conservation laws as well as an integral formula for corresponding conserved densities. The action of a symmetry (discrete or continuous) on a conservation law yields conservation laws. Conservation laws yield non-locally related systems that, in turn, can yield nonlocal symmetries and in addition be useful for the application of other mathematical methods. From its admitted symmetries or multipliers for conservation laws, one can determine whether or not a given system of differential equations can be linearized by an invertible transformation.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Linearization through symmetries for discrete equations

We show that one can define through the symmetry approach a procedure to check the linearizability of a difference equation via a point or a discrete Cole-Hopf transformation. If the equation is linearizable the symmetry provides the linearizing transformation. At the end we present few examples of applications for equations defined on four lattice points

Algorithms for Mappings and Symmetries of Differential Equations

Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points