Constraints of the D$\Delta$KP hierarchy to the semi-discrete AKNS and Burgers hierarchies

TL;DR

This paper explores eigenfunction constraints on the DΔKP hierarchy, revealing connections to semi-discrete AKNS and Burgers hierarchies through algebraic structures and symmetries.

Contribution

It introduces new eigenfunction constraints for the DΔKP system, linking it to semi-discrete AKNS and Burgers hierarchies with algebraic proofs.

Findings

Revisits squared eigenfunction symmetry constraint of DΔKP.

Introduces linear eigenfunction constraint leading to sdBurgers hierarchy.

Establishes equivalence of constraints via recursive algebraic structures.

Abstract

The paper investigates three eigenfunction constraints of two (2+1)-dimensional differential-difference integrable systems. First, we revisit the known squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (D$\Delta$KP) hierarchy, which gives rise to a semi-discrete Ablowitz-Kaup-Newell-Segur hierarchy. Second, we introduce a linear eigenfunction constraint for the D$\Delta$KP system and obtain a combined semi-discrete Burgers (sdBurgers) hierarchy. In the third one, we consider another linear eigenfunction constraint for the modified D$\Delta$KP system and obtain the same combined sdBurgers hierarchy. All these constraint results are proved by using recursive algebraic structures of the involved integrable hierarchies generated by their master symmetries.