Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 050, 22 pages      arXiv:2410.21878      https://doi.org/10.3842/SIGMA.2025.050

Bäcklund-Darboux Transformations for Super KdV Type Equations

Lingling Xue a, Shasha Wang a and Qing Ping Liu b
a) Department of Applied Mathematics, Ningbo University, Ningbo 315211, P.R. China
b) Department of Mathematics, China University of Mining and Technology, Beijing 100083, P.R. China

Received October 30, 2024, in final form June 26, 2025; Published online July 03, 2025

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 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 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.

Key words: Darboux transformations; Bäcklund transformations; Miura transformations; super KdV equations.

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