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The Extended Bigraded Toda hierarchy

Guido Carlet
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Abstract

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 $ε$-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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Guido Carlet. 2008-11-04. The Extended Bigraded Toda hierarchy. https://doi.org/10.1088/0305-4470/39/30/003

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