The Extended Bigraded Toda hierarchy

Details The Extended Bigraded Toda hierarchy The Extended Bigraded Toda hierarchy

2006-04-11

arxiv.org/abs/math-ph/0604024v2

J. Phys. A: Math. Gen. 39 (2006) 9411-9435

Physics

Mathematical Physics

32 pages, corrected typos

Scientific paper

10.1088/0305-4470/39/30/003

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series.

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