Defects in the supersymmetric mKdV hierarchy via Backlund transformations

The integrability of the ${\cal N}=1$ supersymmetric modified Korteweg de-Vries (smKdV) hierarchy in the presence of defects is investigated through the construction of its super B"acklund transformation. The construction of such transformation is performed by using essentially two methods: the B"acklund-defect matrix approach and the superfield approach. Firstly, we employ the defect matrix associated to the hierarchy which turns out to be the same for the supersymmetric sinh-Gordon (sshG) model. The method is general for all flows and as an example we derive explicitly the B"acklund equations in components for the first few flows of the hierarchy, namely $t_3$ and $t_5$. Secondly, the supersymmetric extension of the B"acklund transformation in the superspace formalism is constructed for those flows. Finally, this super B"acklund transformation is employed to introduce type I defects for the supersymmetric mKdV hierarchy. Further integrability aspects by considering modified conserved quantities are derived from the defect matrix.

💡 Research Summary

The paper investigates the integrability of the N = 1 supersymmetric modified Korteweg‑de Vries (smKdV) hierarchy when type‑I defects are present. Using two complementary approaches—a defect‑matrix (or gauge‑matrix) method and a superspace (superfield) formulation—the authors construct a universal super‑Bäcklund transformation that applies to every flow of the hierarchy.

First, the authors recall that the smKdV hierarchy can be generated from the affine super‑Lie algebra b sl(2, 1). The zero‑curvature condition