Dressing method and the coupled KP hierarchy

Abstract

The coupled KP hierarchy, introduced by Hirota and Ohta, are investigated by using the dressing method. It is shown that the coupled KP hierarchy can be reformulated as a reduced case of the 2-component KP hierarchy.

New integrable hierarchies from vertex operator representations of polynomial Lie algebras

“…2 ), giving Lie algebraic explanation of what done in the recent literature [16]. Therefore the soliton solution for the coupled KdV hierarchies can be recovered from those written for the coupled KP equations (5.15) erasing the even variables.…”

“…The more significative consequence of the equations (4.11) are the following commutation relations: 16) which can be checked as follows (4.16) is due to the fact that can be used to define a representation of a "polynomial" generalisation gl ∞ be the infinite dimensional Lie algebra given by the tensor product gl…”

“…Our starting point is the following "coupled KdV equations" which appears in many very recent papers of different authors like Hirota, Hu, Tang, [10], Sakovich [22], Kakei [16]: v t + 6vv x + v xxx = 0 w t + 6vw x + w xxx = 0.…”

Pfaff $\tau$ -functions

“…18 As noted in p. , X treatsτ as a vector (τ n ) n∈A padded with zeros, i.e.,τ n ≡ 0 for n odd. So χ (z) appearing in X(z) acts onτ n as multiplication by z n , and Λ acts onτ as (Λ τ ) n =τ n−1 .…”

“…The Pfaff lattice appears implicitly in the work of Jimbo and Miwa as one half of the D ∞ -hierarchy (compare ( 0.7) (or (3.2)) with the case l = l of formula (7.7) in [15]), in the work of Hirota et al, in the context of the coupled KP hierarchy (compare, e.g., (0.5) and (0.9) with formulas (3.5) and (3.25a) in [13], respectively), in the work of Kac and van de Leur [16] in the context of the DKP hierarchy (on the exact connection, see forthcoming work by J. van de Leur [24]), and in the recent work of S. Kakei [17,18], who realized Hirota et al's coupled KP hierarchy as a restriction of the 2-component KP hierarchy instead of the 2-Toda lattice, and studied its relation to matrix integrals among other aspects.…”