Squared eigenfunction symmetry of the D$\Delta$mKP hierarchy and its constraint

TL;DR

This paper explores the squared eigenfunction symmetry of the D$ ext{ extDelta}$mKP hierarchy, establishing connections to the R-Toda, Burgers, and Volterra hierarchies, and introduces reductions and transformations linking these integrable systems.

Contribution

It introduces new symmetry constraints for the D$ ext{ extDelta}$mKP hierarchy and reveals novel reductions to well-known integrable hierarchies, including the R-Toda and Volterra systems.

Findings

The positive R-Toda hierarchy arises from the symmetry constraint.

An invertible transformation links positive and negative R-Toda hierarchies.

A one-field reduction of the Ragnisco-Tu hierarchy to the Volterra hierarchy is found.

Abstract

In this paper squared eigenfunction symmetry of the differential-difference modified Kadomtsev-Petviashvili (D$\Delta$mKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the D$\Delta$mKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R-Toda) hierarchy. An invertible transformation is given to connect the positive and negative R-Toda hierarchies. The positive R-Toda hierarchy is reduced to the differential-difference Burgers hierarchy. We also consider another D$\Delta$mKP hierarchy and show that its squared eigenfunction symmetry constraint gives rise to the Volterra hierarchy. In addition, we revisit the Ragnisco-Tu hierarchy which is a squared eigenfunction symmetry constraint of the differential-difference Kadomtsev-Petviashvili (D$\Delta$KP) system. It was thought the Ragnisco-Tu hierarchy does not exist one-field reduction, but here we find an one-field reduction to reduce the hierarchy to the Volterra hierarchy. Besides, the differential-difference Burgers hierarchy are also investigated in Appendix. A multi-dimensionally consistent 3-point discrete Burgers equation is given.