New reductions of the Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies via symmetry constraints

Abstract

New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy and the two-dimensional Toda lattice (2DTL) hierarchy are considered on the basis of Sato’s approach. Within this approach these hierarchies are represented by infinite sets of equations for potentials u1,u2,u3,..., of pseudodifferential operators and their eigenfunctions ψ and adjoint eigenfunctions ψ*. The KP and the 2DTL hierarchies are studied under constraints of the following type: ∑n=1N αnSn(u1,u2,u3,...)=Ωx, where Sn are symmetries for these hierarchies, αn are arbitrary constants, and Ω is an arbitrary linear functional of the quantity ψ(λ)ψ*(μ). It is shown that for the KP hierarchy these constraints give rise to hierarchies of 1+1-dimensional commuting flows for the variables u2,u3,...,uN,ψ,ψ*. Many known systems and several new ones are among them. Symmetry reductions for the 2DTL hierarchy give rise both to finite-dimensional dynamical systems and 1+1-dimensional discrete systems. Some few results for the modified KP hierarchy are also presented.