CDIFF: a REDUCE package for computations in geometry of differential equations

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The Reduce package CDIFF was developed with the aim of computing symmetries and conservation laws for PDEs. Its core features are two: defining vector fields on a supermanifold (ie, with even and odd variables) and integrating large and sparse systems of linear algebraic equations.

The development of the package started in the early '90s at the University of Twente by the following researchers: P. Gragert, Paul Kersten, G. Post, M. Roelofs. CDIFF has been recently included in the official REDUCE distribution on sourceforge. It can run on Linux and Windows, both 32-bit and 64-bit, as well as Mac OS.

The name CDIFF was chosen much after the package was developed, precisely when it was included in the Reduce source tree in 2010. It comes from the fact that the package allows to do computations with differential operators in total derivatives, i.e. C-differential operators, for differential equations in even and odd variables.

The package CDIFF was used by P. Kersten, J.S. Krasil'shchik and A.M. Verbovetsky, later joined by R. Vitolo, in the years 2000-2010 for a series of papers aimed at finding integrability related structures like Hamiltonian, symplectic and recursion operator by a method that was published in the paper:

P. Kersten, I. Krasil'shchik, and A. Verbovetsky, Hamiltonian operators and -coverings, J. Geom. Phys. 50 (2004), 273-302, http://arxiv.org/abs/math/0304245.

The CDIFF package consists of three main source files and of a few auxiliary files. They can be downloaded here:

  • Supervf.red - The Super vectorfield package for REDUCE 3.4, Revision: 1.1
  • Tools21.red - The Algebraic operator tools for REDUCE 3.6, Version 2.1 (August 14, 1996)

The following papers describe the Twente CDIFF packages:

G.H.M. Roelofs, The SUPER VECTORFIELD package for REDUCE. Version 1.0, Memorandum 1099, Dept. Appl. Math., University of Twente, 1992

G.H.M. Roelofs, The INTEGRATOR package for REDUCE. Version 1.0, Memorandum 1100, Dept. Appl. Math., University of Twente, 1992

G.F. Post, A manual for the package TOOLS 2.1, Memorandum 1331, Dept. Appl. Math., University of Twente, 1996

Raffaele Vitolo wrote a user guide to CDIFF. It explains:

  • how to install and use REDUCE+CDIFF in Linux and in Windows;
  • how to write CDIFF programs for various types of computations like: higher symmetries and conservation laws, local and nonlocal Hamiltonian operators, etc.

Examples discussed in the user guide are also available. In the user guide examples of computations are performed for evolutionary PDEs.

The new Reduce package CDE simplifies many computations that can be performed by CDIFF, and introduces many new features. Users are encouraged to have a look at the CDE manual.

Recently, a book has been published on symbolic computations with CDE and CDIFF, with a lot of examples discussed in detail. Please have a look here.