New hierarchies of integrable lattice equations and associated properties: Darboux transformation, conservation laws and integrable coupling

In this paper, squared eigenfunction symmetry of the differential‐difference modified Kadomtsev–Petviashvili (DnormalΔmKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the DnormalΔmKP 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 DnormalΔmKP 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 (DnormalΔKP) system. It was thought the Ragnisco–Tu hierarchy did not exist one‐field reduction, but here we find a one‐field reduction to reduce the hierarchy to the Volterra hierarchy. Besides, the differential‐difference Burgers hierarchy is also investigated in the Appendix. A multidimensionally consistent three‐point discrete Burgers equation is given.

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“…[46]) to study the R-Toda lattice; Equation (4.18) was also restudied in Ref. [47] without mentioning the connection with the R-Toda lattice.…”

Section: 2mentioning

confidence: 99%

Squared eigenfunction symmetry of the DmKP hierarchy and its constraint

Chen

Zhang

2021

Stud Appl Math

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“…has been used as an unusual spectral problem (see Eq. (2.1) in [26]) to study the R-Toda lattice; (4.18) was also restudied in [50] without mentioning the connection with the R-Toda lattice.…”

Section: Symmetry Constraint and The R-toda Hierarchymentioning

confidence: 99%

On the squared eigenfunction symmetry of the Toda lattice hierarchy

Cheng

2013

Journal of Mathematical Physics

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Abstract. The squared eigenfunction symmetry for the Toda lattice hierarchy is explicitly constructed in the form of the Kronecker product of the vector eigenfunction and the vector adjoint eigenfunction, which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively. Then after the Fay-like identities and some important relations about the wave functions are investigated, the action of the squared eigenfunction related to the additional symmetry on the tau function is derived, which is equivalent to the Adler-Shiota-van Moerbeke (ASvM) formulas.

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“…when we take u = (u n , v n ) T , and [5][6][7][8]. The bilinear form and Casorati determinant solution for the RTL (1.6) were given in [9].…”

Section: Introductionmentioning

confidence: 99%