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

Abstract

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.

28 pages, LaTeX

References in corpus (2)

• Dispersionful analogues of Benney's equations and $N$-wave systems
• The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies

Cited by in corpus (9)

• Compatible Poisson brackets of hydrodynamic type
• Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures
• Frobenius submanifolds
• Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems
• On the Benney Hierarchy: free energy, string equation and quantization
• Recursion Operator and Rational Lax Representation
• Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics
• Recursion Operators of Some Equations of Hydrodynamic Type
• Darboux-Egoroff Metrics, Rational Landau-Ginzburg Potentials and the Painleve VI Equation