On Lax equations of the two-component BKP hierarchy

A KP-mKP hierarchy via pseudo-differential operators with two derivations

“…From these Bäcklund transformations one recovers the gauge transformations between the KP and the mKP hierarchies studied in [17]. It can also be seen that, if we take the eigenfunction e β = 1, and assume Φ * ν = ∂ ν Φ −1 ν ∂ −1 ν for ν ∈ {1, 2}, then the KP-mKP hierarchy with k ∈ Z odd + is reduced to the two-component BKP hierarchy, which is deeply related to representation theory and algebraic geometry [4,5,9,14,18].…”

“…  e ξ(t2;z) , (5.10) Proof. The proof is similar with that of theorem 3.9 in [9]. Firstly, we introduce the following set of indices…”

“…Let us recall the notion of pseudo-differential operators containing two derivations, following [9]. Let B be a commutative associative algebra of smooth complex functions of two variables x and y.…”

“…>0 are well defined. Such reduced flows compose the two component BKP hierarchy; see [9] and references therein.…”

“…Then the KP-mKP hierarchy will be seen equivalent to the following bilinear equation (see theorem 5.6 below) Res z z −1 w 1 (t 1 , t 2 ; z) w † 1 (t 1 , t 2 ; z) = Res z z −1 w 2 (t 1 , t 2 ; z) w † 2 (t 1 , t 2 ; z) (1.6) for arbitrary time variables (t 1 , t 2 ) and (t 1 , t 2 ). With a method similar as that in [7][8][9], we will define two tau functions τ 1 (t 1 , t 2 ) and τ 2 (t 1 , t 2 ) of the KP-mKP hierarchy. Furthermore, we will see that these tau functions are related to the (adjoint) Baker-Akhiezer functions via the formulae:…”