Lie discrete symmetries of lattice equations

We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the discrete symmetries of the discrete Painlev\'e I equation and of the Toda lattice equation

Algorithmic Determination Of Structure Of Infinite Lie Pseudogroups Of Symmetries Of PDEs

We describe a method which uses a finite number of differentiations and linear operations to determine the Cartan structure coefficients of a structurally transitive Lie pseudogroup from its infinitesimal defining equations. If the defining system is of first order and the pseudogroup has no scalar invariants, the structure coefficients can be simply extracted from the coefficients of the infinitesimal system. We give an algorithm which reduces the higher order case to the first order case. The reduction process uses only differentiation and linear eliminations, for which several well-known algorithms are available. Our method makes feasible the calculation of the Cartan structure of infinite Lie pseudogroups of symmetries of differential equations. Examples including the KP equation and Liouville's equation are given. 1 INTRODUCTION This paper is one of a series in which we investigate the determination of structure of infinite Lie pseudogroups. The main results from the preprint [1..

Algorithmic Determination Of Commutation Relations For Lie Symmetry Algebras Of PDEs

We present an algorithm Commutation Relations, which can calculate the commutation relations for the Lie symmetry algebra of symmetry operators for any system of PDEs. Unlike existing methods, Commutation Relations does not depend on the heuristic process of integrating the associated dierential equations for the symmetry operators (i.e. integrating the `determining equations'). An algorithm Initial Data, developed in previous work, is used to calculate lists of initial data which are in 1-to-1 correspondence with solutions of determining equations. Commutation Relations exploits this correspondence by calculating commutators in terms of initial data. The method has been implemented in the symbolic language Maple and can be applied to both nite- and innite-dimensional Lie symmetry algebras. We show how knowledge of the Lie symmetry algebra calculated by Commutation Relations can simplify the task of explicitly integrating determining equations. 1. INTRODUCTION Consider a system o..