A REDUCE package for finding conserved densities of systems of implicit difference?difference equations*1

Abstract

A computer algebra program for finding polynomial conserved densities of implicit difference–difference equations is presented. The algorithm is based on scaling properties and implemented in computer algebra system REDUCE. The package is applicable to systems of any number of nonlinear difference–difference equations of polynomial type. Title of program: TXCD Catalogue identifier: ADTS Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTS Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computers: PC/AT compatible machine Operating systems: Windows 2000 Programming language used: REDUCE 3.6, RLISP Memory required to execute with typical data: Depends on the problem, minimum about 2 M bytes. No. of bits in a word: 32 No. of bytes in distributed program, including test data, etc.: 10 005 No. of lines in distributed program, including test data, etc.: 1739 Distribution format: tar gzip file Nature of physical problem: The existence of conserved densities for difference–difference equations is of interest for their classification and for understanding the stability of their solutions. Restriction on the complexity of the problem: The program can handle difference–difference equations which can be transformed to polynomial ones, and determine the homogeneous conservation laws. Typical running time: It depends on the equation and the rank of the conserved density. It increases exponentially with the rank of the conserved density. The running times on the PC Pentium with operating systems Windows 2000 (Xeon, 1.7 GHz) are shown in the table below. Timings are given in milliseconds. Performance on Windows Example Rank 0 1 2 3 4 5 6 7 8 9 1(i) 15 15 15 31 150 718 5483 28176 127914 493092 1(ii) 15 15 16 63 170 2264 11171 103938 299001 1386468 1(iii) 15 15 15 46 250 5686 29210 203190 924372 1(iv) 15 15 15 31 47 156 1031 8905 40485 194595 2 15 15 45 187 2358 36673 433794 3(i) 15 63 1780 66518 1390030 ∗∗ 3(ii) 15 47 829 37640 786594 ∗∗ The cases ∗∗ were rejected by memory error.