Squared eigenfunction symmetry of the DmKP hierarchy and its constraint

Abstract

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.

Symmetries of the DΔmKP hierarchy and their continuum limits

“…Since in the recent paper, 12 the squared eigenfunction symmetry constraint leads the DΔmKP system to the spectral problem and the positive hierarchy of the relativistic Toda, not those of the DNLS as we expected, our particular interest in the current paper is to find a new formulation of the squared eigenfunction symmetry constraint so that the constrained DΔmKP system matches the continuous results described in Section 3. In addition, the match of the algebraic structures in the continuum limit is also important because this means that the discretization keeps well the integrable structures.…”

“…In a recent paper, 12 the squared eigenfunction symmetry constraint for the differential‐difference modified KP (DΔmKP) hierarchy was investigated. The later is related to the spectral problem $$$$\begin{equation} \bar{L}\bar{\phi }_n =\lambda \bar{\phi }_n,\nobreakspace \nobreakspace \bar{L}=\bar{v} \Delta +\bar{v}_0 + \bar{v}_1 \Delta ^{-1} + \bar{v}_2 \Delta ^{-2} +\cdots .$$…”

“…7), which is also known as the Darboux transformation of the AKNS spectral problem, 10 but it is different from the familiar discretization, namely, the Ablowitz-Ladik spectral problem. 11 In a recent paper, 12 the squared eigenfunction symmetry constraint for the differentialdifference modified KP (DΔmKP) hierarchy was investigated. The later is related to the spectral problem L φ𝑛 = 𝜆 φ𝑛 , L = vΔ + v0 + v1 Δ −1 + v2 Δ −2 + ⋯ .…”

On the Fourth-Order Lattice Gel'fand-Dikii Equations

“…For the quadrilateral ABS equations [1] and 3-point discrete Burgers equation [7,42], both of which are CAC, they can be obtained as compatibilities of their Lax pairs constructed from the MDC property (see [6,7,42]). However, for a DBSQ type equation, e.g., (B-2) or ((A-2), (C-3)), its Lax pairs, no matter constructed from the DL scheme or by means of the MDC property, are incomplete in the sense that the original DBSQ equation can not be fully recovered from the compatibility of its Lax pair (see [22,44]).…”

On the fourth-order lattice Gel'fand-Dikii equations

“…For the quadrilateral ABS equations [13] and 3-point discrete Burgers equation [35,36], both of which are CAC, they can be obtained as compatibilities of their Lax pairs constructed from the MDC (see [34][35][36]. However, for a DBSQ type equation, e.g (B-2) or ((A-2), (C-3)), its Lax pairs, no matter constructed from the DL scheme or by means of the MDC property, are incomplete in the sense that the original DBSQ equation can not be fully recovered from the compatibility of its Lax pair (see [20]).…”