Vertex operator solutions to the discrete KP-hierarchy

Abstract

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.

32 pages

References in corpus (1)

• Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems

Cited by in corpus (24)

• Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy
• Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems
• Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems
• Combinatorics of dispersionless integrable systems and universality in random matrix theory
• Multivariate orthogonal polynomial and integrable systems
• Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies
• Functional representations of integrable hierarchies
• The multicomponent 2D Toda hierarchy: Discrete flows and string equations
• Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy
• The full Kostant-Toda hierarchy on the positive flag variety
• Bilinear equations in Darboux transformations by Boson-Fermion correspondence
• Nonassociativity and Integrable Hierarchies
• Extension of the discrete KP hierarchy
• Quantum curves
• Rational Ruijsenaars-Schneider hierarchy and bispectral difference operators
• Another Christoffel--Darboux Formula for Multiple Orthogonal Polynomials of Mixed Type
• Semiclassical expansions in the Toda hierarchy and the hermitian matrix model
• Local and non-local multiplicative Poisson vertex algebras and differential-difference equations
• Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies
• Heat kernel expansions on the integers and the Toda lattice hierarchy
• n-th discrete KP hierarchy
• On a relationship between high rank cases and rank one cases of Hermitian random matrix models with external source
• The extended D-Toda hierarchy
• Addition formulae of discrete KP, q-KP and two-component BKP systems