Integrability properties of the differential-difference Kadomtsev–Petviashvili hierarchy and continuum limits

Abstract

The paper reveals clear links between the differential-difference Kadomtsev–Petviashvili hierarchy and the (continuous) Kadomtsev–Petviashvili hierarchy. Isospectral differential-difference Kadomtsev–Petviashvili flows and non-isospectral differential-difference Kadomtsev–Petviashvili flows are derived through Lax triad approach. The Lax triads also provide simple zero-curvature representations for the obtained flows. The non-isospectral flow acts as a master symmetry to provide recursive relations for the obtained flows. These flows generate a Lie algebra, which is a starting point for investigating more integrability properties. We derive symmetries, Hamiltonians and conserved quantities for the isospectral differential-difference Kadomtsev–Petviashvili hierarchy. The Lie algebras generated respectively by the flows, symmetries, Hamiltonians and conserved quantities have same structures. Finally, we provide a uniform continuum limit which is different from Miwa's transformation. By means of defining degrees of some elements w.r.t. the continuum limit, we prove that in the uniform continuum limit the differential-difference Kadomtsev–Petviashvili hierarchies together with their Lax triads, zero-curvature representations and integrability characteristics go to their continuous counterparts. Structure deformation of Lie algebras in the continuum limit is also explained.