B\"acklund-Darboux Transformations for Super KdV Type Equations

TL;DR

This paper develops Bäcklund-Darboux transformations for super KdV equations, deriving new hierarchies, reductions, and exact solutions, thereby unifying and extending previous results in the field.

Contribution

It introduces a unified framework for Bäcklund-Darboux transformations for super KdV equations, including new hierarchies, reductions, and explicit solutions, connecting various known equations.

Findings

Derived generalized super mKdV equation from super KdV via Miura transformation.

Constructed Bäcklund-Darboux transformations for multiple super KdV equations.

Obtained exact solutions and unified the transformations across hierarchies.

Abstract

By introducing a Miura transformation, we derive a generalized super modified Korteweg-de Vries (gsmKdV) equation from the generalized super KdV (gsKdV) equation. It is demonstrated that, while the gsKdV equation takes Kupershmidt's super KdV (sKdV) equation and Geng-Wu's sKdV equation as two distinct reductions, there are also two equations, namely Kupershmidt's super modified KdV (smKdV) equation and Hu's smKdV equation, which are associated with the gsmKdV equation. By analyzing the flows within the gsKdV and gsmKdV hierarchies, we specifically derive the first negative flows associated with both hierarchies.We then construct a number of B\"acklund-Darboux transformations (BDTs) for both the gsKdV and gsmKdV equations, elucidating the interrelationship between them. By proper reductions, we are able not only to recover the previously known BDTs for Kupershimdt's sKdV and smKdV equations, but also to obtain the BDTs for the Geng-Wu's sKdV/smKdV and Hu's smKdV equations. As applications, we construct some exact solutions for those equations. Since all flows of the sKdV or smKdV hierarchy share the same spatial parts of spectral problem, thus these Darboux matrices and spatial parts of BTs are applicable to any flow of those hierarchies.