A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briey outlined. Examples illustrate the use of the software. Directions for further research and development are indicated.

Classical and nonclassical symmetries of the nonlinear heat equation ut = uxx + f(u), (1) are considered. The method of differential Gröbner bases is used both to find the conditions on f(u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f(u) in terms of the roots of f(u) =0. 0 Symmetry Reductions of a Nonlinear Heat Equation 1

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1+3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalization of the well-known symmetry reductions of the nonlinear wave equations.

We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasi-exactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.

After reviewing some notions of the formal theory of differential equations we discuss the completion of a given system to an involutive one. As applications to symmetry theory we study the effects of local solvability and of gauge symmetries, respectively. We consider non-classical symmetry reductions and more general reductions using differential constraints.

An analytical study, strongly aided by computer algebra packages diffgrob2 by Mansfield and rif by Reid, is made of the 3 + 1-coupled nonlinear Schrodinger (CNLS) system i\Psi t +r 2 \Psi + i j\Psij 2 + j\Phij 2 j \Psi = 0; i\Phi t +r 2 \Phi + i j\Psij 2 + j\Phij 2 j \Phi = 0: This system describes transverse effects in nonlinear optical systems. It also arises in the study of the transmission of coupled wave packets and "optical solitons", in nonlinear optical fibres. First we apply Lie's method for calculating the classical Lie algebra of vector fields generating symmetries that leave invariant the set of solutions of the CNLS system. The large linear classical determining system of PDE for the Lie algebra is automatically generated and reduced to a standard form by the rif algorithm, then solved, yielding a 15-dimensional classical Lie invariance algebra. A generalization of Lie's classical method, called the nonclassical method of Bluman and Cole, is applied to th...

We present an informal overview of a number of approaches to differential equations which are popular in computer algebra. This includes symmetry and completion theory, local analysis, differential ideal and Galois theory, dynamical systems and numerical analysis. A large bibliography is provided. 1 Introduction Differential equations represent one of the largest fields within mathematics. Besides being an interesting subject of their own right one can hardly overestimate their importance for applications. They appear in natural and engineering sciences and increasingly often in economics and social sciences. Whenever a continuous process is modeled mathematically, chances are high that differential equations are used. Thus it is not surprising that differential equations also play an important role in computer algebra and most general purpose computer algebra systems provide some kind of solve command. Many casual users believe that designing and improving such procedures is a centra...

We give the exposition of a generalized symmetry approach to reduction of initial value problems for nonlinear evolution equations in one spatial variable. Using this approach we classify the initial value problems for third-order evolution equations that admit reduction to Cauchy problems for systems of two ordinary di#erential equations. These reductions are shown to correspond to higher conditional symmetries admitted by the corresponding nonlinear evolution equations.

A model ‘remarkable ’ fin equation is singled out from a class of nonlinear (1 + 1)-dimensional fin equations. For this equation a number of exact solutions are constructed by means of using both classical Lie algorithm and different modern techniques (functional separation of variables, generalized conditional symmetries, hidden symmetries etc). 1

We classify initial-value problems for a class of one-dimensional evolution equations, which can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The technique applied relies heavily on higher conditional symmetries of the equations under study, which means, in particular, that the obtained reductions cannot be derived within the framework of the standard Lie approach.