Symmetries of Partial Differential Equations Conservation Laws — Applications — Algorithms

2 The authors of these issues involve not only mathematicians, but also speci­ alists in (mathematical) physics and computer sciences. So here the reader will find different points of view and approaches to the considered field. A. M. VINOGRADOV 3 Acta Applicandae Mathematicae 15: 3-21, 1989. © 1989 Kluwer Academic Publishers. Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results A. M. VINOORADOV Department of Mathematics, Moscow State University, 117234, Moscow, U. S. S. R. (Received: 22 August 1988) Abstract. The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed. AMS subject classifications (1980). 35A30, 58005, 58035, 58H05. Key words. Higher symmetries, conservation laws, partial differential equations, infinitely prolonged equations, generating functions. o. Introduction In this paper we present the basic notions and results from the general theory of local symmetries and conservation laws of partial differential equations. More exactly, we will focus our attention on the main conceptual points as well as on the problem of how to find all higher symmetries and conservation laws for a given system of partial differential equations. Also, some general views and perspectives will be discussed.

I (Acta Appl. Math. 15, 1–210).- Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results.- Symmetries and Conservation Laws of Kadomtsev-Pogutse Equations (Their computation and first applications).- Symmetries and Conservation Laws of Navier-Stokes Equations.- Symmetries, Invariant Solutions and Conservation Laws of the Nonlinear Acoustics Equation.- Local Nonintegrability of Long-Short Wave Interaction Equations.- On Symmetries and Conservation Laws of the Equations of Shallow Water with an Axisymmetric Profile of Bottom.- On Symmetries of the Heat Equation.- Nonlocal Trends in the Geometry of Differential Equations: Symmetries, Conservation Laws, and Bäcklund Transformations.- II (Acta Appl Math. 16, 1–142).- Exactly and Completely Integrable Nonlinear Dynamical Systems.- Recursion and Group Structures of Soliton Equations.- Building of Mathematical Models of Continuum Media on the Basis of the Invariance Principle.- III (Acta Appl. Math. 16,143–242).- Efficiently Implementing Two Methods of the Geometrical Theory of Differential Equations: An Experience in Algorithm and Software Design.- Computations in Differential and Difference Modules.- Reducing Systems of Linear Differential Equations to a Passive Form.- Software to Compute Infinitesimal Symmetries of Exterior Differential Systems, with Applications.- Lie Algebra Computations.