Discrete Integrable Models Research Articles

Multi-pole soliton of discrete integrable equations and modified Riemann-Hilbert approach: discrete Hirota equation

In this study, the Riemann-Hilbert approach was developed and applied to the discrete Hirota equation. We constructed a modified Riemann-Hilbert problem compatible with the discrete Hirota equation and derived a reconstruction formula for its solutions. Because the characteristic function contains a potential, we modify the Riemann-Hilbert approach to make the Riemann-Hilbert matrix have good asymptotic properties. We believe that the modified Riemann-Hilbert approach can also be applied to other discrete integrable models. By using the direct method of Laurent series, we obtained the expression of multi-pole solutions for the discrete Hirota equation and demonstrated the dynamic behavior of some solutions.

Bi-infinite Solutions for KdV- and Toda-Type Discrete Integrable Systems Based on Path Encodings

We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical ‘Pitman’s transformation’ of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural ‘carrier process’ for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.

THE QUANTUM H3 INTEGRABLE SYSTEM

The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H3-invariants "polynomially"-isospectral to the quantum H3 model is defined.

Algebra-geometric constructions of a (2+1)-dimensional discrete integrable model

A new (2+1)-dimensional differential-difference equation is considered. With the aid of the nonlinearization of the Lax pair, the (2+1)-dimensional differential-difference equation is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems. Based on the theory of algebra curve, the continuous flow and discrete flow are straightened out in view of the introduced Abel–Jacobi coordinates.

Quantum deformed magnon kinematics

The dispersion relation for planar N=4 supersymmetric Yang-Mills is identified with the Casimir of a quantum deformed two-dimensional kinematical symmetry, E_q(1,1). The quantum deformed symmetry algebra is generated by the momentum, energy and boost, with deformation parameter q=e^{2\pi i/\lambda}. Representing the boost as the infinitesimal generator for translations on the rapidity space leads to an elliptic uniformization with crossing transformations implemented through translations by the elliptic half-periods. This quantum deformed algebra can be interpreted as the kinematical symmetry of a discrete integrable model with lattice spacing given by the BMN length a=2\pi/\sqrt{\lambda}. The interpretation of the boost generator as the corner transfer matrix is briefly discussed.

A novel approach to the theory of Padé approximants

By associating polynomials and power series expansions with modules we describe the theory of Padé approximants in terms of tensor products of representations and interpret their recurrence relations algebraically. The treatment links with the theory of Hirota derivatives and discrete integrable models and is generalizable to larger numbers of variables. This paper summarizes and extends the work of [2].

Equations of motion and conserved quantities in non-Abelian discrete integrable models

Conserved quantities for the Hirota bilinear difference equation, which is satisfied by eigenvalues of the transfer matrix, are studied. The transfer-matrix eigenvalue combinations that are integrals of motion for discrete integrable models, which correspond to Ak−1 algebras and satisfy zero or quasi-periodic boundary conditions, are found. Discrete equations of motion for a non-Abelian generalization of the Liouville model and the discrete analogue of the Tsitseiko equation are obtained.

Hamiltonian structures of the multi-boson KP hierarchies, abelianization and lattice formulation

We present a new form of the multi-boson reduction of KP hierarchy with Lax operator written in terms of boson fields abelianizing the second Hamiltonian structure. This extends the classical Miura transformation and the Kupershmidt-Wilson theorem from the (m) KdV to the KP case. A remarkable relationship is uncovered between the higher Hamiltonian structures and the corresponding Miura transformations of KP hierarchy, on one hand, and the discrete integrable models living on refinements of the original lattice connected with the underlying multi-matrix models, on the other hand. For the second KP Hamiltonian structure, worked out in details, this amounts to finding a series of representations of the nonlinear Ŵ ∞ algebra in terms of arbitrary finite number of canonical pairs of free fields.