Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations

Abstract

The bi-Hamiltonian structure of certain multicomponent integrable systems, generalizations of the dispersionless Toda hierarchy, is studied for systems derived from a rational Lax function. One consequence of having a rational rather than a polynomial Lax function is that the corresponding bi-Hamiltonian structures are degenerate, i.e., the metric that defines the Hamiltonian structure has a vanishing determinant. Frobenius manifolds provide a natural setting in which to study the bi-Hamiltonian structure of certain classes of hydrodynamic systems. Some ideas on how this structure may be extended to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, is given.