Generalized Bigraded Toda Hierarchy

Bigraded Toda hierarchy L₁M(n)=L₂N(n) is generalized to L₁M(n)=L₂N(n)+∑j∈ Z∑ᵢ₌₁ᵐq⁽ⁱ⁾ₙΛʲr⁽ⁱ⁾ₙ₊₁, which is the analogue of the famous constrained KP hierarchy Lᵏ= (Lᵏ)≥0+∑ᵢ₌₁ᵐqᵢ∂⁻¹rᵢ. It is known that different bosonizations of fermionic KP hierarchy will give rise to different kinds of integrable hierarchies. Starting from the fermionic form of constrained KP hierarchy, bilinear equation of this generalized bigraded Toda hierarchy (GBTH) are derived by using 2--component boson--fermion correspondence. Next based upon this, the Lax structure of GBTH is obtained. Conversely, we also derive bilinear equation of GBTH from the corresponding Lax structure.