Symbolic Representation and Classification of Supersymmetric Evolutionary Equations

Abstract

We extend the symbolic representation to the ring of N=1 supersymmetric differential polynomials, and demonstrate that operations on the ring, such as the super derivative, Fréchet derivative, and super commutator, can be carried out in the symbolic way. Using the symbolic representation, we classify scalar λ‐homogeneous N=1 supersymmetric evolutionary equations with nonzero linear term when λ>0 for arbitrary order and give a comprehensive description of all such integrable equations.

On the (Non)Removability of Spectral Parameters in Z 2 $\mathbb{Z}_{2}$ -Graded Zero-Curvature Representations and Its Applications

“…Furthermore, using the results described in Sect. 4 and techniques from [31], we prove the integrability of new fifth order N = 1 supersymmetric evolution equation (9a)-(9b) found by Tian and Liu (see [32,33]): in Example 7 we construct its sl(9|8)-valued zero-curvature representation with nonremovable parameter.…”

“…Example 3 Using the technique described in [23], let us examine an sl(2|1)-valued zerocurvature representation for the equation which found by Tian and Liu (Case F in [32], see also [33]):…”

Casting light on shadow Somos sequences

“…We have not explicitly described the analogues of these maps and their solutions over D, but such a description is a straightforward consequence of our results on the dual Somos-4. Continuous integrable systems with Grassmann variables have been studied for several decades, with one of the most recent results being a symmetry classification of N = 1 supersymmetric scalar homogeneous evolutionary PDEs [26]. Many of our considerations here extend naturally to the dual number analogues of other discrete integrable systems, such as the family of maps in [13], which are connected with Gale-Robinson sequences and cluster algebras, or the higher genus analogues of (3.4) in [12].…”

Supersymmetric Sawada-Kotera Equation: Bäcklund-Darboux Transformations and Applications

“…which reduces to (1) when the fermionic variable ξ is set to zero. It is mentioned that the SSK equation also appears in the symmetry classification of supersymmetric integrable systems [42]. As its classical counterpart, the SSK equation is also integrable and its integrability is ensured by presenting a Lax representation, the existence of infinitely many conserved quantities and a recursion operator [41].…”

“…which reduces to (1) when the fermionic variable ξ is set to zero. It is mentioned that the SSK equation also appears in the symmetry classification of supersymmetric integrable systems [42].…”