Sato–Bäcklund transformations and string equations of the mKP hierarchy

Abstract

Additional symmetry is an important kind of symmetries depending explicitly on the time and space variables, which can be expressed through Sato–Bäcklund transformations. In this paper, we construct Sato–Bäcklund transformations of the modified KP hierarchy and its constrained cases. Then the string equations of the [Formula: see text]-reduced modified KP hierarchy are established by requiring the system independent on some additional symmetry flows, which are expressed by the Lax operator [Formula: see text] and the Orlov–Shulman’s operator [Formula: see text]. At last, we obtain the negative Virasoro constraint on the two tau functions of the 2-reduced modified KP hierarchy satisfying the string equations.

Bilinear equations in Darboux transformations by Boson-Fermion correspondence

“…Then e H(t) ψ(λ)e −H(t) = e ξ(t,λ) ψ(λ), e H(t) ψ * (λ)e −H(t) = e −ξ(t,λ) ψ * (λ), (9) where ξ(t, λ) = ∞ n=1 t n λ n . If introduce the Bosonic Fock space B = C[z, z −1 , t 1 , t 2 , t 3 , • • • ], then there exists an isomorphism σ t : F −→ B given by σ t (a|0 ) = j z j j|e H(t) a|0 , a ∈ A.…”

“…Remark: Note that [9] Res ∂ L n = (log τ 0 ) xtn and Res ∂ (∂L n ∂ −1 ) * = −(log τ 1 ) xtn , which can be obtained by comparing the ∂ 0 and ∂ −1 -terms in the evolution equation (26) of the dressing operator Z. Thus each v i in L can be expressed by (log τ 0 ) xtn and v 0 = (log(τ 1 /τ 0 )) x , or (log τ 1 ) xtn and v 0 .…”