Hirota Bilinear Difference Equation Research Articles

DISCRETE SOLITONS IN INFINITE REDUCED WORDS

We consider a discrete dynamical system where the roles of the states and the carrier are played by translations in an affine Weyl group of type A. The Coxeter generators are enriched by parameters, and the interactions with the carrier are realized using Lusztig’s braid move (a, b, c) ↦ (bc/(a+c), a+c, ab/(a+c)). We use wiring diagrams on a cylinder to interpret chamber variables as τ-functions. This allows us to realize our systems as reductions of the Hirota bilinear difference equation and thus obtain N-soliton solutions.

Solitons of a simple nonlinear model on the cubic lattice

We study a simple nonlinear model defined on the cubic lattice. We propose a bilinearization scheme for the field equations and demonstrate that the resulting system is closely related to the well-studied integrable models, such as the Hirota bilinear difference equation and the Ablowitz–Ladik system. This result is used to derive the two sets of the N-soliton solutions.

Solitons of a vector model on the honeycomb lattice

We study a simple nonlinear vector model defined on the honeycomb lattice. We propose a bilinearization scheme for the field equations and demonstrate that the resulting system is closely related to the well-studied integrable models, such as the Hirota bilinear difference equation and the Ablowitz-Ladik system. This result is used to derive the N-soliton solutions.

Explicit solutions for a nonlinear model on the honeycomb and triangular lattices

We study a simple nonlinear model defined on the honeycomb and triangular lattices. We propose a bilin-earization scheme for the field equations and demonstrate that the resulting system is closely related to the well-studied integrable models, such as the Hirota bilinear difference equation and the Ablowitz-Ladik system. This result is used to derive the two sets of explicit solutions: the N-soliton solutions and ones constructed of the Toeplitz determinants.

Double-dimers, the Ising model and the hexahedron recurrence

We define and study a recurrence relation in Z3, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition sums for edge configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with certain double-dimer configurations of a graph. We compute limit shapes for the corresponding double-dimer configurations.The Kashaev difference equation arising in the Ising model Y-Delta relation is a special case of the hexahedron recurrence. In particular this reveals the cluster nature underlying the Ising model. The above relation allows us to prove a Laurent phenomenon for the Kashaev difference equation.

Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics

We show that eigenvalues of the family of Baxter Q-operators for supersymmetric integrable spin chains constructed with the g l ( K | M ) -invariant R-matrix obey the Hirota bilinear difference equation. The nested Bethe ansatz for super-spin chains, with any choice of simple root system, is then treated as a discrete dynamical system for zeros of polynomial solutions to the Hirota equation. Our basic tool is a chain of Bäcklund transformations for the Hirota equation connecting quantum transfer matrices. This approach also provides a systematic way to derive the complete set of generalized Baxter equations for super-spin chains.

When is negativity not a problem for the ultradiscrete limit?

The “ultradiscrete limit” has provided a link between integrable difference equations and cellular automata displaying soliton-like solutions. In particular, this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the “Max” operator. Although it certainly is the case that dropping these positivity conditions creates potential difficulties, it is still possible for solutions to persist under the ultradiscrete limit, even in their absence. To recognize when this will occur, one must consider whether a certain expression, involving a measure of the rates of convergence of different terms in the difference equation and their coefficients, is equal to zero. Applications discussed include the solution of elementary ordinary difference equations, a discretization of the Hirota Bilinear Difference Equation and the identification of integrals of motion for ultradiscrete equations.

Tau-functions, Grassmannians and rank one conditions

In integrable systems, specifically the KP hierarchy, there are functions known as “tau-functions”, closely related to the Schur polynomials in terms of which they are often written. Although they are generally viewed as the solutions to a collection of nonlinear PDEs, in this note they will equivalently be characterized by a quadratic difference equation. Sato's theorem associates tau-functions to the points of a Grassmann manifold. To make that amazing theorem clear to non-experts, we will first show an analogous (but easily understood) example of a linear ODE and its solution from a flow on the xy-plane. In each case the solution is created via a flow generated by a certain linear operator. The question we pose is this: “What other operators could have been used to generate solutions in the same way?” Although the answer is well known in the ODE case, the question in the nonlinear case is the main result of our new paper. We will state the result and discuss its relationship to the “trend” of writing tau-functions in terms of matrices satisfying certain rank one conditions. The elucidation of a geometric interpretation of the Hirota bilinear difference equation (HBDE) is a key feature of the proof and will be briefly described.

On KP generators and the geometry of the HBDE

Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the “shift” operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including the Hirota bilinear difference equation (HBDE). Here, we rederive the HBDE as a statement about linear maps between Grassmannians. In addition to illustrating the fundamental nature of this equation in the standard theory, we make use of this geometric interpretation of the HBDE to answer the question of what other infinitesimal generators could be used for similarly creating tau-functions. The answer to this question involves a “rank one condition”, tying this investigation to the existing results on integrable systems involving such conditions and providing an interpretation for their significance in terms of the relationship between the HBDE and the geometry of Grassmannians.

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.

An extension of the Hirota bilinear difference equation

An extension of the Hirota bilinear difference equation to a multilinear, multidimensional lattice space is discussed. This extension admits linear Backlund transformations. A preliminary result on solutions is presented in the case of trilinear equations.

Hirota equation and Bethe ansatz

Recent analyses of classical integrable structures in quantum integrable models solved by various versions of the Bethe ansatz are reviewed. Similarities between elements of quantum and classical theories of integrable systems are discussed. Some key ideas in quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions in classical soliton theory. Functional relations for quantum transfer matrices become the classical Hirota bilinear difference equation; solving this classical equation gives all the basic results for the spectral properties of quantum systems. Vice versa, typical Bethe ansatz formulas under certain boundary conditions yield solutions of this classical equation. The Baxter T-Q relation and its generalizations arise as auxiliary linear problems for the Hirota equation.

A symmetric generalization of linear Bäcklund transformation associated with the Hirota bilinear difference equation

The Hirota bilinear difference equation is generalized to discrete space of arbitrary dimension. Solutions to the nonlinear difference equations can be obtained via B\"acklund transformation of the corresponding linear problems.

Hirota’s difference equations

A review of selected topics in Hirota's bilinear difference equation (HBDE) is given. This famous 3-dimensional difference equation is known to provide a canonical integrable discretization for most important types of soliton equations. Similarly to the continuous theory, HBDE is a member of an infinite hierarchy. The central point of our exposition is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to 2-dimensional equations are considered. Among them discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain and other typical examples are discussed in detail.

Discrete Hirota's Equation in Quantum Integrable Models

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.

Quantum Integrable Models and Discrete Classical Hirota Equations

The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A k-1 -type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived.

Integrable difference analogue of the logistic equation and Bäcklund transformation of the KP hierarchy

A difference analogue of the logistic equation, which preserves integrability, is derived from Hirota's bilinear difference equation. The integrability of the map is shown to result from the large symmetry associated with the Bäcklund transformation of the KP hierarchy. We introduce a scheme which interpolates between this map and the standard logistic map and enables us to study integrable and nonintegrable systems on an equal basis. In particular we study the behaviour of Julia set at the point where the nonintegrable map passes to the integrable map.

Lax formalism of discrete time integrablw systems

We present a Lax form for integrable systems with a discrete time variable. In particular we study Hirota's bilinear difference equation, which is one of the representations of the KP hierarchy. The Lax form of the Toda lattice is obtained from our formalism in a suitable continuous limit of independent variables. We also obtain an equation of motion for the monodromy matrix based on the discrete-variable Lax form considering quasi-periodic solutions.

The Virasoro constraint and Hirota's bilinear difference equation

From the perspective of Hirota's bilinear difference equation, the connection of the tau function and the conformal symmetry is shown to become apparent and the Virasoro constraint is naturally derived. In this formalism, the compatibility of the flow provided by conformal symmetry and the KP flow is manifest.

Integrable lattice gauge model based on soliton theory

A gauge model Lagrangian on a three-dimensional lattice is proposed. Under certain conditions equations of motion are reduced to the Backlund transformation of Hirota's bilinear difference equation, which is solved by every solution of the Kadomtsev-Petviashvili hierarchy. Generation of soliton solutions given in the Casorati determinant form is discussed explicitly.