Negative Flows of the potential KP-hierarchy

Abstract

Abstract. We construct a Grassmannian-like formulation for the potential KP-hierarchy including additional "negative" flows. Our approach will generalize the notion of a τ -function to include negative flows. We compare the resulting hierarchy with results by Hirota, Satsuma and Bogoyavlenskii.

The Bilinear negative Kadomtsev–Petviashvili system and its Kac–Moody–Virasoro symmetry group

“…Negative flows of the KP system may have other kinds of extensions such as the Grassmannian description [18] and two-component extensions [19].…”

“…Actually, the bilinear form given by Hu et al [20] is corresponding to q(y, t) = 2, i.e., equations ( 5) and ( 6) of [30]. On the other hand, if one takes y = x, q = 0 and/or q = 1 in (18) and (19), it is just two known bilinear forms of the classical Bossinesq equation [24]. In this paper, we focus our attention on the q = 0 case for simplicity.…”

On a Negative Flow of the AKNS Hierarchy and Its Relation to a Two-Component Camassa-Holm Equation

“…A standard example is provided by the modified KdV-hierarchy, which can be embedded in a new extended hierarchy. This extended hierarchy contains in addition to the original modified KdV equation also the differential equation of the sine-Gordon model realized as the first negative flow [2,3,4,5,6,7].…”

Weakly Nonassociative Algebras, Riccati and KP Hierarchies

“…Besides their appearance in control and systems theory, matrix Riccati equations (see[19][20][21], for example) frequently showed up in the context of integrable systems, see in particular[22][23][24][25][26][27][28][29][30][31][32][33][34].3 f has to be differentiable, of course, which requires a corresponding (e.g. Banach space) structure on A.…”