A new extended matrix KP hierarchy and its solutions

With the square eigenfunctions symmetry constraint, we introduce a new extended matrix KP hierarchy and its Lax representation from the matrix KP hierarchy by adding a new $\tau_B$ flow. The extended KP hierarchy contains two time series ${t_A}$ and ${\tau_B}$ and eigenfunctions and adjoint eigenfunctions as components. The extended matrix KP hierarchy and its $t_A$-reduction and $\tau_B$ reduction include two types of matrix KP hierarchy with self-consistent sources and two types of (1+1)-dimensional reduced matrix KP hierarchy with self-consistent sources. In particular, the first type and second type of the 2+1 AKNS equation and the Davey-Stewartson equation with self-consistent sources are deduced from the extended matrix KP hierarchy. The generalized dressing approach for solving the extended matrix KP hierarchy is proposed and some solutions are presented. The soliton solutions of two types of 2+1-dimensional AKNS equation with self-consistent sources and two types of Davey-Stewartson equation with self-consistent sources are studied.Comment: 17 page

Vertex operator solutions to the discrete KP-hierarchy

Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite sequence obtained by Darboux transforming an arbitrary KP solution recursively forward and backwards, yields a solution to the discrete KP-hierarchy. The latter is a KP hierarchy where the continuous space x-variable gets replaced by a discrete n-variable. The fact that these sequences satisfy the discrete KP hierarchy is tantamount to certain bilinear relations connecting the consecutive KP solutions in the sequence. At the Grassmannian level, these relations are equivalent to a very simple fact, which is the nesting of the associated infinite-dimensional planes (flag). It turns out that many new and old systems lead to such discrete (semi-infinite) solutions, like sequences of soliton solutions, with more and more solitons, sequences of Calogero-Moser systems, having more and more particles, band matrices, etc... ; this will be developped in another paper. In this paper, as an other example, we show that the q-KP hierarchy maps, via a kind of Fourier transform, into the discrete KP hierarchy, enabling us to write down a very large class of solutions to the q-KP hierarchy.Comment: 32 page

Morphological evidence for enhanced kisspeptin and neurokinin B signaling in the infundibular nucleus of the aging man.

Peptidergic neurons synthesizing kisspeptin (KP) and neurokinin B (NKB) in the hypothalamic infundibular nucleus have been implicated in negative sex steroid feedback to GnRH neurons. In laboratory rodents, testosterone decreases KP and NKB expression in this region. In the present study, we addressed the hypothesis that the weakening of this inhibitory testosterone feedback in elderly men coincides with enhanced KP and NKB signaling in the infundibular nucleus. This central hypothesis was tested in a series of immunohistochemical studies on hypothalamic sections of male human individuals that were divided into arbitrary "young" (21-49 yr, n = 11) and "aged" (50-67 yr, n = 9) groups. Quantitative immunohistochemical experiments established that the regional densities of NKB-immunoreactive (IR) perikarya and fibers, and the incidence of afferent contacts they formed onto GnRH neurons, exceeded several times those of the KP-IR elements. Robust aging-dependent enhancements were identified in the regional densities of KP-IR perikarya and fibers and the incidence of afferent contacts they established onto GnRH neurons. The abundance of NKB-IR perikarya, fibers, and axonal appositions to GnRH neurons also increased with age, albeit to lower extents. In dual-immunofluorescent studies, the incidence of KP-IR NKB perikarya increased from 36% in young to 68% in aged men. Collectively, these immunohistochemical data suggest an aging-related robust enhancement in central KP signaling and a moderate enhancement in central NKB signaling. These changes are compatible with a reduced testosterone negative feedback to KP and NKB neurons. The heavier KP and NKB inputs to GnRH neurons in aged, compared with young, men may play a role in the enhanced central stimulation of the reproductive axis. It requires clarification to what extent the enhanced KP and NKB signaling upstream from GnRH neurons is an adaptive response to hypogonadism or, alternatively, a consequence of a decline in the androgen sensitivity of KP and NKB neurons

Construction of KP Hierarchies in Terms of Finite Number of Fields and their Abelianization

The $2M$-boson representations of KP hierarchy are constructed in terms of $M$ mutually independent two-boson KP representations for arbitrary number $M$. Our construction establishes the multi-boson representations of KP hierarchy as consistent Poisson reductions of standard KP hierarchy within the $R$-matrix scheme. As a byproduct we obtain a complete description of any finitely-many-field formulation of KP hierarchy in terms of Darboux coordinates with respect to the first Hamiltonian structure. This results in a series of representations of \Win1\, algebra made out of arbitrary even number of boson fields.Comment: 12 p., LaTeX, minor typos corrected, BGU-93/2/June-P