Bi-Hamiltonian formulation of the Kadomtsev–Petviashvili and Benjamin–Ono equations

Abstract

It was shown recently that the Kadomtsev–Petviashvili (KP) equation (an integrable equation in 2+1, i.e., in two-spatial and one-temporal dimensions) admits a bi-Hamiltonian formulation. This was achieved by considering KP as a reduction of a (3+1)-dimensional system (in the variables x,y1, y2,t). It is shown here, using the KP as a concrete example, that equations in 2+1 possess two bi-Hamiltonian formulations and two recursion operators. Both Hamiltonian operators associated with the x direction are local; in contrast only one of the Hamiltonian operators associated with the y direction is local. Furthermore, using the Benjamin–Ono equation as a concrete example, it is shown that intergrodifferential equations in 1+1 admit an algebraic formulation analogous to that of equations in 2+1.