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International Journal of Modern Physics A

DISCRETE KAUP–NEWELL EQUATION, CANONICAL BÄCKLUND TRANSFORMATION AND BAXTER'S Q-OPERATOR

A. Ghose Choudhury, Asesh Roy Chowdhury

Published Feb 10, 2002• 2 citations
Publisher

Abstract

We consider the Bäcklund transformation (BT) for a discretized version of the integrable Kaup–Newell equation, obtained recently by Tsuchida through a generalization of the Ablowitz–Ladik equation. The canonical nature of the classical Bäcklund transformation is proved by determining a suitable generating function. The corresponding Q-operator is determined by means of a quantization procedure proposed by Sklyanin et al. The latter's connection with Baxter's t - Q relation is explored and a comparison with the corresponding relation for the Bethe eigenvalue is made by solving the quantum inverse problem.

DOI
10.1142/s0217751x02006043

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We consider the Bäcklund transformation (BT) for a discretized version of the integrable Kaup–Newell equation, obtained recently by Tsuchida through a generalization of the Ablowitz–Ladik equation. The canonical nature of the classical Bäcklund transformation is proved by determining a suitable generating function. The corresponding Q-operator is determined by means of a quantization procedure proposed by Sklyanin et al. The latter's connection with Baxter's t - Q relation is explored and a comparison with the corresponding relation for the Bethe eigenvalue is made by solving the quantum inverse problem.
We construct the Baxter operator and the corresponding Baxter equation for a quantum version of the Ablowitz–Ladik model. The result is achieved in two different ways: by using the well-known Bethe ansatz technique and by looking at the quantum analogue of the classical Bäcklund transformations. General results about integrable models governed by the same r -matrix algebra will be given. Baxter's equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Bäcklund transformations gives a trace formula representing the classical analogue of Baxter's equation. A q-integral representation of the Baxter operator is discussed.
Abstract We derive the quantum analogue of a Bäcklund transformation for the quantized Ablowitz–Ladik chain, a space discretization of the nonlinear Schrödinger equation. The quantization of the Ablowitz–Ladik chain leads to the q -boson model. Using a previous construction of Baxter’s Q -operator for this model by the author, a set of functional relations is obtained which matches the relations of a one-variable classical Bäcklund transform to all orders in . We construct also a second Q -operator and show that it is closely related to the inverse of the first. The multi-Bäcklund transforms generated from the Q -operator define the fusion matrices of a 2D TQFT and we derive a linear system for the solution to the quantum Bäcklund relations in terms of the TQFT fusion coefficients.
On Bäcklund transformations and boundary conditions associated with the quantum inverse problem for a discrete nonlinear integrable system and its connection to Baxter’s QQQ-operator
A discrete nonlinear system is analysed in case of open chain boundary conditions at the ends.It is shown that the integrability of the system remains intact, by obtaining a modified set of Lax equations which automatically take care of the boundary conditions.The same Lax pair also conforms to the conditions stipulated by Sklyanin [5].The quantum inverse problem is set up and the diagonalisation is carried out by the method of sparation of variables.Backlund transformations are then derived under the modified boundary conditions using the classical r-matrix .Finally by quantising the Backlund transformation it is possible to identify the relation satisfied by the eigenvalue of Baxter's Q-operator even for the quasi periodic situation.
We explore the notion of the quantum auxiliary linear problem and the associated problem of quantum Backlund transformations (BT). In this context we systematically construct the analogue of the classical formula that provides the whole hierarchy of the time components of Lax pairs at the quantum level for both closed and open integrable lattice models. The generic time evolution operator formula is particularly interesting and novel at the quantum level when dealing with systems with open boundary conditions. In the same frame we show that the reflection KKK-matrix can also be viewed as a particular type of BT, fixed at the boundaries of the system. The qqq-oscillator (qqq-boson) model, a variant of the Ablowitz-Ladik model, is then employed as a paradigm to illustrate the method. Particular emphasis is given to the time part of the quantum BT as possible connections and applications to the problem of quantum quenches as well as the time evolution of local quantum impurities are evident. A discussion on the use of Bethe states as well as coherent states and the path integral formulation for the study of the time evolution is also presented.
The recent progress in revealing classical integrable structures in quantum models solved by Bethe ansatz is reviewed. Fusion relations for eigenvalues of quantum transfer matrices can be written in the form of classical Hirota’s bilinear difference equation. This equation is also known as the completely discretized version of the 2D Toda lattice. We explain how one obtains the specific quantum results by solving the classical equation. The auxiliary linear problem for the Hirota equation is shown to generalize Baxter’s T-Q relation.