New negative grade solitonic sector for supersymmetric KdV and mKdV hierarchies

TL;DR

This paper develops a systematic approach to construct supersymmetric negative graded flows for mKdV and KdV hierarchies using $sl(2,1)$ algebra, revealing new solutions and classifications based on vacuum states.

Contribution

It introduces a gauge super Miura transformation linking smKdV and sKdV, classifies flows by vacuum solutions, and constructs new soliton solutions with deformed vertex operators.

Findings

Mapped smKdV and sKdV via gauge super Miura transformation.

Classified flows according to vacuum solutions.

Constructed new solutions with non-zero bosonic and fermionic vacua.

Abstract

A systematic construction for supersymmetric negative graded (non-local) flows for mKdV and KdV based on $sl(2,1)$ with a principal gradation is proposed in this paper. We show that smKdV and sKdV can be mapped onto each other through a gauge super Miura transformation, together with an additional condition for the negative flows, which ensure the supersymmetry of the negative sKdV flow. In addition, we classify both smKdV and sKdV flows with respect to the vacuum (boundary) solutions. These are classified according to zero or non-zero vacuum. Each vacuum solution is used to derive both soliton solutions and the corresponding Heisenberg subalgebra for the smKdV hierarchy. We present the new solutions corresponding to non-zero bosonic and fermionic vacuum by constructing the deformed vertex operators. Finally, the gauge Miura transformation is employed to obtain the sKdV solutions, which exhibit a rich degeneracy due to both multiple gauge super Miura transformations and multiple vacuum possibilities.