Bäcklund-Darboux Transformations for Super KdV Type Equations - Xue, Lingling; Wang, Shasha; Liu, Qing Ping (Symmetry, Integrability and Geometry: Methods and Applications, 2025)
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 several Bäcklund-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 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 the spectral problem, these Darboux matrices and spatial parts of BTs apply to any flow of those hierarchies.