Bilinear identities for the constrained modified KP hierarchy

Abstract

In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ , the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ .

A new generalized constrained modified KP hierarchy

“…When one choose c is zero, the c-k gcmKP hierarchy will become the usual constrained mKP hierarchy. 6,19,24,27 Proposition 7. The definition of c-k gcmKP hierarchy is reasonable; that is, (1), (18), and ( 19) are consistent with each other.…”

“…14 By now, much work has been done in the aspects of the constrained KP and mKP hierarchies, for example, additional Viraroso symmetries, 6,21,22 Hamiltonian structures, 12,15,23 Darboux transformations, [24][25][26] and bilinear formulations. 16,17,27 Here in this paper, we generalize the constrained mKP hierarchy from (L k ) ≤0 = q𝜕 −1 r into a more general case by adding a multiple of the inverse of Lax operator in the nonpositive part, that is, (L k ) ≤0 = q𝜕 −1 r + cL −1 , which is called the generalized constrained mKP (gcmKP for short) hierarchy. In this new constraint, the constant c makes the new system become more complicated, which is helpful when considering the asymptotic value of q and r.…”

Bilinear equations in Darboux transformations by Boson-Fermion correspondence

“…Note that if charge of a = l, then the terms j = l on the right hand side will be zero according to the Wick Theorem. By using the isomorphism σ t and the following formulas [28,41], l|ψ(λ)e H(t) = λ l−1 l − 1|e H(t−ε(λ −1 )) , l|ψ * (λ)e H(t) = λ −l l + 1|e H(t+ε(λ −1 )) , (10) with ε(λ −1 ) = (λ −1 , λ −2 /2, λ −3 /3, • • • ), one can realize the Fermions ψ i and ψ * i in the forms of the Bosonic operators [28,41]…”

“…which can be viewed as the analogue of (85) in the modified KP hierarchy. ( 93) is also very important in the proof of the ASvM formula in the mKP case and derivation of the bilinear equations of the l-constrained modified KP hierarchy: L l = (L l ) ≥1 +Φ∂ −1 Ψ∂ (one can refer to [10,13] for these results).…”

“…At last, some new examples of the bilinear relations are given. Particularly, the bilinear equations involving the binary Darboux transformations are very important in the proof of the Adler-Shiota-van Moerbeke (ASvM) formulas in the additional symmetries [1,13,17,34,49], and in the derivation of the bilinear equations [10,15,35,47] of the symmetry constraints of the KP, modified KP and BKP hierarchies. This paper is organized in the way below.…”