Symmetries of the DΔmKP hierarchy and their continuum limits

Abstract

In the recent paper [Stud. App. Math. 147 (2021), 752], squared eigenfunction symmetry constraint of the differential‐difference modified Kadomtsev–Petviashvili (DΔmKP) hierarchy converts the DΔmKP system to the relativistic Toda spectral problem and its hierarchy. In this paper, we introduce a new formulation of independent variables in the squared eigenfunction symmetry constraint, under which the DΔmKP system gives rise to the discrete spectral problem and a hierarchy of the differential‐difference derivative nonlinear Schrödinger equation of the Chen–Lee–Liu type. In addition, by introducing nonisospectral flows, two sets of symmetries of the DΔmKP hierarchy and their algebraic structure are obtained. We then present a unified continuum limit scheme, by which we achieve the correspondence of the mKP and the DΔmKP hierarchies and their integrable structures.

On Symmetries of Integrable Quadrilateral Equations

“…with (20) as their zero-curvature representations. The Lax pair of the above hierarchy now is composed of (11a) and…”

“…from the zero-curvature representations (20), we can derive that the isospectral flows {K (l) } satisfy (cf. [22,25])…”

“…In the study of symmetries of continuous and differential-difference equations with Lax pairs, the zero-curvature representations of isospectral and non-isospectral flows have proven to be a powerful tool in deriving symmetries and their Lie algebras for the isospectral equation hierarchy as well as non-isospectral equation hierarchy, e.g., [15][16][17][18][19][20][21][22][23][24][25][26]. In this paper, we aim to extend this method to Lax-integrable P∆Es, with a particular focus on isospectral flows.…”