Equivalence of two constructions for $\widehat{sl}_2$-integrable hierarchies

We discuss the equivalence between the Date–Jimbo–Kashiwara–Miwa (DJKM) construction and the Kac–Wakimoto (KW) construction of $\widehat{sl}_2$-integrable hierarchies within the framework of bilinear equations. The DJKM method has achieved remarkable success in constructing integrable hierarchies associated with classical A, B, C, D affine Lie algebras. In contrast, the KW method exhibits broader applicability, as it can be employed even for exceptional E, F, G affine Lie algebras. However, a significant drawback of the KW construction lies in the great difficulty of obtaining Lax equations for the corresponding integrable hierarchies. Conversely, in the DJKM construction, Lax structures for numerous integrable hierarchies can be derived. The derivation of Lax equations from bilinear equations in the KW construction remains an open problem. Consequently, demonstrating the equivalent DJKM construction for the integrable hierarchies obtained via the KW construction would be highly beneficial for obtaining the corresponding Lax structures. In this paper, we use the language of lattice vertex algebras to establish the equivalence between the DJKM and KW methods in the $\widehat{sl}_2$-integrable hierarchy for principal and homogeneous representations.