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

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

1998-07-08

arxiv.org/abs/solv-int/9807004v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

28 pages, LaTeX

Scientific paper

10.1063/1.533015

The bi-Hamiltonian structure of certain multi-component integrable systems, generalizations of the dispersionless Toda hierarchy, is studies 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 which defines the Hamiltonian structure has 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 extanded to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, are given.

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