ABSTRACT
At a very informal but practically convenient level, we discuss the step/by/step computation of nonlocal recursions for symmetry algebras of nonlinear coupled boson/fermion N = 1 $ N=1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math5253.tif"/> supersymmetric systems by using the SsTools environment. The principle of symmetry plays an important role in modern mathematical physics. The differential equations that constitute integrable models practically always admit symmetry transformations. The presence of symmetry transformation in a system yields two types of explicit solutions: those which are invariant under a transformation (sub)group and the solutions obtained by propagating a known solution by the same group. The recursion operator is a (pseudo)differential operator which maps symmetries of a given system into symmetries of the same system. The recursion operators allow obtaining new symmetries for a given seed symmetry. It is common for important equations of mathematical physics not to have local recursion operators other than the identity id : φ ↦ φ $ \text{ id}:\varphi \mapsto \varphi $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math5254.tif"/> . Instead, they often admit nonlocal recursions which involve integrations such as taking the inverse of the total derivative D x $ D_x $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math5255.tif"/> with respect to the independent variable x. To describe such nonlocal structures we use the approach of nonlocalities. By nonlocalities we mean an extension of the initial system by new fields such that the initial fields are differential consequences of the new ones. In the case of recursion operators such fields often arise from conservation laws. We refer to a recursion operator for the Korteweg–de Vries equations as a motivating example of a nonlocal recursion operator; see Example 7 on page 400. The supersymmetric integrable systems, i.e. systems involving commuting (bosonic, or even) and anticommuting (fermionic, or odd) independent variables and/or unknown functions, have found remarkable applications in modern mathematical physics (for example supergravity models, perturbed conformal field theory [12]; we refer to [1, 4] for a general overview). When dealing with supersymmetric models of theoretical physics, it is often hard to predict whether a certain mathematical 391approximation will be truly integrable or not. Therefore we apply the symbolic computation to exhibit necessary integrability features. In what follows we restrict ourselves to the case of N = 1 $ N=1 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math5256.tif"/> (where N refers to the number of odd anticommuting independent variables θ i $ \theta _i $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429470462/301849f2-53e7-4646-acba-6259bf733f01/content/inline-math5257.tif"/> ). Nevertheless, the techniques and computer programs described below could be easily applied to the case of arbitrary N. Usually, the N is not bigger than 8; see for example [11] and [3]. It is an interesting open problem to establish criteria that set a limit on N in “N-extended” supersymmetric equations of mathematical physics. The latest version of SsTools can be found at [16]; see also [8]. We refer to [1, 4] and [2, 5, 6, 9, 13] for reviews of the geometry and supergeometry of partial differential equations. We refer to [10] for an overview of other software that could be used for similar computational tasks.
