Tau function and Hirota bilinear equations for the Extended bigraded Toda Hierarchy

Abstract

In this paper we generalize the Sato theory to the extended bigraded Toda hierarchy (EBTH). We revise the definition of the Lax equations,give the Sato equations, wave operators, Hirota bilinear identities (HBI) and show the existence of $tau$ function $\tau (t)$. Meanwhile we prove the validity of its Fay-like identities and Hirota bilinear equations (HBEs) in terms of vertex operators whose coefficients take values in the algebra of differential operators. In contrast with HBEs of the usual integrable system, the current HBEs are equations of product of operators involving $e^{{\partial }_x}$ and $\tau (t)$.

29 pages, to appear Journal of Mathematical Physics(2010)

References in corpus (5)

• Integrable Hierarchies and Dispersionless Limit
• The Topological CP^1 Model and the Large-N Matrix Integral
• Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy
• The Extended Bigraded Toda hierarchy
• Fay-like identities of the Toda Lattice Hierarchy and its dispersionless limit

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• On the extended multi-component Toda hierarchy
• Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of CP^1 orbifolds
• Quantum Torus symmetry of the KP, KdV and BKP hierarchies
• Block (or Hamiltonian) Lie symmetry of dispersionless D type Drinfeld-Sokolov hierarchy
• Sato theory on the $q$-Toda hierarchy and its extension
• Quantum torus algebras and B(C) type Toda systems
• Bilinear identities for an extended B-type Kadomtsev-Petviashvili hierarchy
• On the squared eigenfunction symmetry of the Toda lattice hierarchy
• Extended lattice Gelfand-Dickey hierarchy
• Constrained lattice-field hierarchies and Toda system with Block symmetry
• Multi-fold Darboux transformations of the extended bigraded Toda hierarchy
• Generalized ILW hierarchy: Solutions and limit to extended lattice GD hierarchy
• A strongly coupled extended Toda hierarchy and its Virasoro symmetry
• Bosonic symmetries of the extended fermionic $(2N,2M)$-Toda hierarchy