312 research outputs found
Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems
Classically, a single weight on an interval of the real line leads to
moments, orthogonal polynomials and tridiagonal matrices. Appropriately
deforming this weight with times t=(t_1,t_2,...), leads to the standard Toda
lattice and tau-functions, expressed as Hermitian matrix integrals.
This paper is concerned with a sequence of t-perturbed weights, rather than
one single weight. This sequence leads to moments, polynomials and a (fuller)
matrix evolving according to the discrete KP-hierarchy. The associated
tau-functions have integral, as well as vertex operator representations.
Among the examples considered, we mention: nested Calogero-Moser systems,
concatenated solitons and m-periodic sequences of weights. The latter lead to
2m+1-band matrices and generalized orthogonal polynomials, also arising in the
context of a Riemann-Hilbert problem.
We show the Riemann-Hilbert factorization is tantamount to the factorization
of the moment matrix into the product of a lower- times upper-triangular
matrix.Comment: 40 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
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