CDE: a Reduce package for integrabilty of PDEs

The Reduce package CDE is written 'on top' of the Reduce package CDIFF. It is devoted to integrability of general differential equations (i.e., not necessarily in evolution form) with an arbitrary number of dependent and independent variables.

It can compute linearization and adjoints of differential operators in total derivatives, generalized symmetries, conservation laws, Hamiltonian, symplectic and recursion operators, with local and nonlocal coordinates. Schouten brackets between local Hamiltonian operators, also in multidimensions, can be computed. The simplecticity tests for local operators can also be computed. The extension of the Schouten bracket to nonlocal operators is currently under development, as well as symplecticity and hereditariety properties for nonlocal operators. It is distributed under the same free license of REDUCE.

CDE is part of the official Reduce distribution, and can be obtained here, click on the more recent snapshot directory. It is distributed in different binary files for various operating systems (Windows, macOS, Linux).

The documentation of CDE is included in the official manual of Reduce; an updated version can be obtained together with the Reduce sources, [1]. See inside the /doc folder. The /doc folder also contains documentation for Reduce programming, and in particular the text Inside Reduce, by A.C. Norman and R. Vitolo.

Examples of computations with CDE are contained in the folder /packages/cde/examples of the Reduce sources.

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.

A new paper has been published on features of CDE:

• R. Vitolo, Computing with Hamiltonian operators, Computer Physics Communication 244 (2019), 228-245, [ArXiv].

Interested users can find [example programs here].

Please feel free to send comments and questions on CDE to the author at raffaele.vitolo@unisalento.it.

The current version of CDE is 3.0. It has been last modified in 05/01/2021 and it is constantly updated. In comparison with the previous version 1.0, starting from version 2.0 CDE is much faster and more reliable, as it is programmed in Reduce symbolic mode, and has an improved user interface. Moreover it has new features and many new example programs.

The old CDE version 1.0 can still be obtained here.