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

TL;DR

This paper demonstrates the equivalence of two different methods for constructing $ olinebreak ext{sl}_2$--integrable hierarchies, which could facilitate deriving Lax equations from bilinear forms and enhance understanding of integrable systems.

Contribution

It proves the equivalence of DJKM and KW constructions for $ olinebreak ext{sl}_2$--integrable hierarchies using lattice vertex algebras, bridging a gap between the two approaches.

Findings

Established the equivalence of DJKM and KW methods for $ ext{sl}_2$ hierarchies.

Provided a framework to derive Lax equations from bilinear equations in KW construction.

Enhanced understanding of integrable hierarchies through algebraic equivalence.

Abstract

In this paper, we investigate the equivalence of Date--Jimbo--Kashiwara--Miwa (DJKM) construction and Kac--Wakimoto (KW) construction for $\widehat{sl}_2$--integrable hierarchies. DJKM method has gained great success in constructions of integrable hierarchies corresponding to classical ABCD affine Lie algebras, while the KW method is more applicable, which can be even used in exceptional EFG affine Lie algebras. But in KW construction, it is quite difficult to obtain Lax equations for the corresponding integrable hierarchies, while in DJKM construction, one can derive Lax structures for many integrable hierarchies. It is still an open problem for the derivation of Lax equations from bilinear equations in KW construction. Therefore if we can show the equivalent DJKM construction for the integrable hierarchies derived by the KW construction, then it will be helpful to get corresponding Lax structures. Here the equivalence of DJKM and KW methods is showed in the $\widehat{sl}_2$--integrable hierarchy for principal and homogeneous representations by using the language of the lattice vertex algebras.