Symbolic computation of the Painlevé test for nonlinear partial differential equations using Maple

Abstract

A software package wkptest written in Maple is presented, which can carry out the traditional Painlevé test for polynomial partial differential equations automatically. Furthermore, some truncated expansions are also obtained whether an equation passes the test or not. The truncated expansions play an important role in studying the integrability, such as Lax pairs, bilinear forms, Darboux transformations and explicit solutions. The effectiveness of wkptest is illustrated by applying it to a variety of equations. Program summary Title of program: wkptest Catalogue identifier: ADTY Program Summary URL: http://cpc.cs.qub.ac.uk/summaries/ADTY Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computers: PC Pentium IV Installations: Copy Operating systems: Windows 98/2000/XP Programming language used: Maple V R6 or higher Memory required to execute with typical data: depends on the problem, minimum about 8M words No. of bits in a word: 8 No. of bytes in distributed program, including test data, etc.: 27 574 No. of lines in distributed program, including test data, etc.: 2500 Distribution format: tar gzip file Nature of physical problem: Our program can perform the traditional Painlevé test for nonlinear partial differential equations automatically, and some truncated expansions are also obtained whether the given equations pass the test. The obtained truncated expansions play an important role in studying the integrability as well as constructing explicit exact solutions. Restriction on the complexity of the problem: The program can handle coupled nonlinear partial differential equations, in which every equation is a polynomial (or can be converted to a polynomial) in the unknown functions and their derivatives. Typical running time: It depends on the input equations, numbers of possible branches and the value of largest resonance. For most of the nonlinear partial differential equations which we have tested, the running time is no more than 10 seconds.