On a Reduction of the Kadomtsev-Petviashvili Hierarchy

This paper analyzes the reduction of the well known Kadomtsev-Petviashvili hierarchy. The reduction yields a previously unknown dispersion counterpart of the dispersionless hierarchy which has a Lax function of the form $p+u(x)(p-\phi {)}^{-1}+v(x)(p-\phi {)}^{-2}$. This paper also describes the bihamiltonian structure of the reduced hierarchy using Dirac reduction and proves that the approximation for the reduced hierarchy up to the second order of the dispersion parameter coincides with the hierarchy of integrable systems constructed from a particular two-dimensional Frobenius manifold using the approach of Dubrovin and Zhang.