Integrability properties of the dispersionless Kadomtsev-Petviashvili hierarchy

Abstract

In the paper we investigate integrability characteristics for the dispersionless KadomtsevPetviashvili hierarchy. These characteristics include symmetries, Hamiltonian structures and conserved quantities. We give a Lax triad to construct a master symmetry and a hierarchy of non-isospectral dispersionless Kadomtsev-Petviashvili flows. These non-isospectral flows, together with the known isospectral dispersionless Kadomtsev-Petviashvili flows, form a Lie algebra, which is used to derive two sets of symmetries for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy. By means of the master symmetry, symmetries, Noether operator and conserved covariants, Hamiltonian structures are constructed for both isospectral and non-isospectral dispersionless Kadomtsev-Petviashvili hierarchies. Finally, two sets of conserved quantities and their Lie algebra are derived for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy.

Optimal System, Invariant Solutions and Conservation Laws of the Dispersionless B type Kadomtsev-Petviashvili Equation

“…Recently, D J Zhang [16] et al investigated the iso and non isospectral flows of dKP hierarchies by introducing Lax triad approach. Many of the interesting properties are yet to be investigated for these types of equations such as underlying Lie algebraic structures, similarity reductions and solutions using Lie point symmetries, integrability properties and solutions via discretization, Painlevé analysis, differential-difference analogues etc.…”