Multicomponent integrable reductions in the Kadomtsev–Petviashvilli hierarchy

Abstract

New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy are considered on the basis of Sato’s approach. Within this approach the KP hierarchy is represented by infinite sets of equations for potentials u₂,u₃,..., of pseudodifferential operators and their eigenfunctions Ψ and adjoint eigenfunctions Ψ*. The KP hierarchy was studied under constraints of the following type (∑ⁿ_i=1 Ψ_iΨ^*_i)_x = S_κ,x where S_κ,x are symmetries for the KP equation and Ψ_i(λ_i), Ψ^*_i(λ_i) are eigenfunctions with eigenvalue λ_i. It is shown that for the first three cases κ=2,3,4 these constraints give rise to hierarchies of 1+1‐dimensional commuting flows for the variables u₂, Ψ₁,...,Ψ_n, Ψ^*₁,...,Ψ^*_n. Bi‐Hamiltonian structures for the new hierarchies are presented.