The solution to the q-KdV equation

Adler, Mark; Horozov, Emil; Moerbeke, Pierre van

doi:10.1016/s0375-9601(98)00082-6

Cited by 36 publications

(57 citation statements)

References 14 publications

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“…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. This was also reported in a brief note with E. Horozov [4].…”

supporting

confidence: 70%

Vertex Operator Solutions to the Discrete KP-Hierarchy

Adler¹,

Moerbeke²

1999

Communications in Mathematical Physics

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127

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Contents 1 The KP τ -functions, Grassmannians and a residue formula 7 2 The existence of a τ -vector and the discrete KP bilinear identity 13 3 Sequences of τ -functions, flags and the discrete KP equation 18 4 Discrete KP-solutions generated by vertex operators 23 5 Example of vertex generated solutions: the q-KP equation 24Vertex operators, which are disguised Darboux maps, transform solutions of the KP equation into new ones. In this paper, we show that the bi-infinite * The final version appeared in: Comm.

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supporting

confidence: 70%

Vertex Operator Solutions to the Discrete KP-Hierarchy

Adler¹,

Moerbeke²

1999

Communications in Mathematical Physics

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127

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“…In [2] and [4], we gave an application of the bi-infinite discrete KP to the q-KP equation. In general, the main features are summarized in the following statement, whose proof can be found in [2]:…”

Section: Aug 24 1998mentioning

confidence: 99%

The Spectrum of Coupled Random Matrices

Adler¹,

Moerbeke²

1999

The Annals of Mathematics

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205

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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 τ -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 KPhierarchy. The associated τ -functions have integral, as well as vertex operator representations. Among the examples considered, we mention: nested Calogero-Moser systems, concatenated solitons and mperiodic sequences of weights. The latter lead to 2m+1-band matrices and generalized orthogonal polynomials, also arising in the context of * Appeared in: Comm. Math. Phys., 207, 589-620 (1999) † Department of Mathematics, Brandeis University, Waltham, Mass 02454, USA. Email: adler@math.brandeis.edu. The support of a National Science Foundation grant # DMS-98-4-50790 is gratefully acknowledged.‡ Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium and Brandeis University, Waltham, Mass 02454, USA. E-mail: vanmoerbeke@geom.ucl.ac.be and @math.brandeis.edu. The support of a National Science Foundation grant # DMS-98-4-50790, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged. Some of the present work was done at the Centre Emile Borel, Paris (fall 96). 1Adler-van Moerbeke:Polyn/RH-problems Aug 24, 1998 §0, p.2 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.

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“…The N -th q-KdV hierarchy becomes q-KdV hierarchy for N = 2. The q-NKdV hierarchy inherited several integrable structures from classical N -th KdV hierarchy, such as infinite conservation laws [10], bi-Hamiltonian structure [11,12], τ function [13,14], Bäcklund transformation [15]. The second type is the q-KP hierarchy [17,22].…”

Section: Introductionmentioning

confidence: 99%

q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

He¹

2006

SIGMA

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Abstract. Using the determinant representation of gauge transformation operator, we have shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP (q-cKP) hierarchy, i.e. l-constraints of q-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of q-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear q-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of q-deformation (q-effects) in single q-soliton from the simplest τ function of the q-KP hierarchy and in multi-q-soliton from one-component q-cKP hierarchy, and their dependence of x and q, were also presented. Finally, we observe that q-soliton tends to the usual soliton of the KP equation when x → 0 and q → 1, simultaneously.

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The solution to the q-KdV equation

Cited by 36 publications

References 14 publications

Vertex Operator Solutions to the Discrete KP-Hierarchy

Vertex Operator Solutions to the Discrete KP-Hierarchy

The Spectrum of Coupled Random Matrices

q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

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