Classification of integrable super-systems using the SsTools environment
Abstract
A classification problem is proposed for supersymmetric evolutionary PDE that satisfy the assumptions of nonlinearity, nondegeneracy, and homogeneity. Four classes of nonlinear coupled boson-fermion systems are discovered under the weighting assumption |f|=|b|=|D|=1/2 >. The syntax of the REDUCE package SSTOOLS, which was used for intermediate computations, and the applicability of its procedures to the calculus of super-PDE are described. Program summaryProgram title:SsTools Catalogue identifier:ADYY_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYY_v1_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:89 178 No. of bytes in distributed program, including test data, etc.:869 212 Distribution format:tar.gz Programming language:REDUCE 3.7, REDUCE 3.8 Computer:(i) IBM PC, (ii) cluster Operating system:LINUX RAM:problem dependent (10 Mb-1 Gb), typical working size <100 Mb Word size:32, 64 bits Nature of problem:The program allows the classification of N⩾1 supersymmetric nonlinear scaling-invariant evolution equations {f=φ,b=φ} that admit infinitely many local symmetries propagated by recursion operators; here b(x,t;θ) is the set of bosonic super-fields and f(x;t;θ) are fermionic super-fields. Solution method:First, (half-)integer weights |f|,|b|,…,|D|,|D|≡1 are assigned to all variables and derivatives and then pairs of commuting flows that are homogeneous w.r.t. these weights are constructed. Secondly, the seeds of higher symmetry sequences [P.J. Olver, Applications of Lie Groups to Differential Equations, second ed., Springer, Berlin, 1993] for the systems are sorted out, and finally the recursion operators that generate the symmetries are obtained [I.S. Krasil'shchik, P.H.M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, Kluwer, Dordrecht, 2000]. The intermediate algebraic systems upon the undetermined coefficients are solved by using [T. Wolf, Applications of Crack in the classification of integrable systems, in: CRM Proc. Lecture Notes, vol. 37, 2004, pp. 283-300]. Restrictions:Computation of symmetries of high differential order for very large evolutionary systems may cause memory restrictions. Additional size/time restrictions may occur if the homogeneity weights of some super-fields are non-positive, see Section 1.2 Unusual features:SsTools has been extensively tested using hundreds of PDE systems within three years on UNIX-based PC-machines. SsTools is applicable to the computation of symmetries, conservation laws, and Hamiltonian structures for N⩾1 evolutionary super-systems with any N. SsTools is also useful for performing extensive arithmetic of general nature including differentiations of super-field expressions. Running time:Depends on the size and complexity of the input system and varies between seconds and minutes. References:[1] http://lie.math.brocku.ca/crack/susy/sstools.red; The support package Crack is obtained from http://lie.math.brocku.ca/crack/src/crack.tar.gz
- Publication:
-
Computer Physics Communications
- Pub Date:
- August 2007
- DOI:
- 10.1016/j.cpc.2007.02.113
- arXiv:
- arXiv:nlin/0609065
- Bibcode:
- 2007CoPhC.177..315K
- Keywords:
-
- 02.30.Ik;
- 02.70.Wz;
- 12.60.Jv;
- Integrable systems;
- Symbolic computation;
- Supersymmetric models;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Mathematical Physics;
- Mathematics - Mathematical Physics
- E-Print:
- MSC 35Q53,37K05,37K10,81T40