Classification of integrable super-systems using the SsTools environment

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|=|Dt|=12. 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:SsToolsCatalogue identifier:ADYY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYY_v1_0.htmlProgram obtainable from:CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.:89 178No. of bytes in distributed program, including test data, etc.:869 212Distribution format:tar.gzProgramming language:REDUCE 3.7, REDUCE 3.8Computer:(i) IBM PC, (ii) clusterOperating system:LINUXRAM:problem dependent (10 Mb–1 Gb), typical working size <100 MbWord size:32, 64 bitsNature of problem:  The program allows the classification of N⩾1N⩾1 supersymmetric nonlinear scaling-invariant evolution equations {ft=φf,bt=φb} that admit infinitely many local symmetries propagated by recursion operators; here b(x,t;θ)b(x,t;θ) is the set of bosonic super-fields and f(x;t;θ)f(x;t;θ) are fermionic super-fields.Solution method:  First, (half-)integer weights |f|,|b|,…,|Dt|,|Dx|≡1|f|,|b|,…,|Dt|,|Dx|≡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 1.2Unusual 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⩾1N⩾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