Analysis of Extended Structures and Construction of Darboux Transformations for Supersymmetric Coupled KdV Systems

Abstract This paper systematically investigates the integrability, extended structures, and exact solutions of a class of coupled Korteweg–de Vries (KdV) models derived from supersymmetric evolution equations. First, the coupled system consisting of bosonic and fermionic field components is obtained by superfield expansion. Within the framework of exterior differential forms, a differentially closed ideal is constructed by employing the method of extended structures. Furthermore, an explicit representation of the Lax pair is derived by using Lie algebra representation theory and the $$\mathfrak {sl}(4,\mathbb {C})$$ algebra, thus establishing the integrability of the system. On this basis, the first-order and higher-order Darboux transformations are systematically derived, which generate explicit expressions for multi-soliton solutions. Finally, exact single- and double-soliton solutions with supersymmetric locking are obtained by taking the zero background as the seed solution. It is demonstrated that the bosonic and fermionic components propagate cooperatively in the form of supermultiplets. This work provides a systematic framework for the algebraic analysis and exact solution construction of supersymmetric integrable systems.