Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are brifley outlined. Examples illustrate the use of the symbolic software.

Contents 1 The purpose of CRACK 2 2 How to apply CRACK 4 2.1 The call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Flags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Contents of the CRACK package 6 3.1 Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Integrating exact PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Direct separation of PDEs . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Indirect separation of PDEs . . . . . . . . . . . . . . . . . . . . . . . 12 3.5 Solving standard ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1 4 Examples 14 4.1 Investigating symmetries of ODEs/PDEs and systems o

The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semi-geostrophic equations in such a way that a well-defined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an ‘infinite linear cascade’. 1.

A method based on infinite parameter conservation laws is described to factor linear differential operators out of nonlinear partial differential equations (PDEs) or out of differential consequences of nonlinear PDEs. This includes a complete linearization to an equivalent linear PDE (-system) if that is possible. Infinite parameter conservation laws can be computed, for example, with the computer algebra package ConLaw.

A survey of techniques and 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 outlined. An exhaustive review of the literature, including old and modern books and papers presenting key concepts is given. Special attention is paid to methods for reducing the determining equations into standard form, and their subsequent integration. Several examples illustrate the use of the Lie symmetry software. Throughout the paper, new trends in the development of symbolic packages for Lie symmetry analysis are indicated.

Contents 1 The purpose of Crack 2 2 Technical details 2 2.1 System requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Updates / web demos . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.4 The files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.5 The call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.6 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.7 Interactive mode, flags, parameters and the list of procedures . . . . . 4 3 Contents of the Crack package 6 3.1 Pseudo Di#erential Grobner Basis . . . . . . . . . . . . . . . . . . . . 7 3.2 Integrating exact PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Direct separation of PDEs . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Indirect separation of PDEs . . . . . . . . . . . . . . . .