Hirota Bilinear Equation Research Articles

EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS

Riemann theta functions are used to construct one-periodic and two-periodic wave solutions to a class of (2+1)-dimensional Hirota bilinear equations. The basis for the involved solution analysis is the Hirota bilinear formulation, and the particular dependence of the equations on independent variables guarantees the existence of one-periodic and two-periodic wave solutions involving an arbitrary purely imaginary Riemann matrix. The resulting theory is applied to two nonlinear equations possessing Hirota bilinear forms: ut + uxxy - 3uuy - 3uxv = 0 and ut + uxxxxy - (5uxxv + 10uxyu - 15u2v)x = 0 where vx = uy, thereby yielding their one-periodic and two-periodic wave solutions describing one-dimensional propagation of waves.

A systematic method to construct Hirota’s transformations of continuous soliton equations and its applications

Hirota’s bilinear method is a powerful tool for obtaining a wide class of exact solutions of soliton equations. The crucial step of this method is to find a suitable dependent variable transformation, i.e. Hirota’s transformation, which transforms a soliton equation into a Hirota bilinear equation. In this paper, a systematic method to construct Hirota’s transformations of continuous soliton equations is proposed. And some examples are given to illuminate the availability of this method. In addition, a new two-soliton solution of a coupled nonlinear Schrödinger equation is obtained.

The complexiton solutions to the nonisospectral Korteweg–de Vries equation

Starting with the solution classification for a linear differential equations, the complexiton solutions to nonisospectral Korteweg–de Vries equation are presented. The basic technique adopted is the Wronskian technique for Hirota's bilinear 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.

Tau functions in combinatorial Bethe ansatz

We introduce ultradiscrete tau functions associated with rigged configurations for A n ( 1 ) . They satisfy an ultradiscrete version of the Hirota bilinear equation and play a role analogous to a corner transfer matrix for the box–ball system. As an application, we establish a piecewise linear formula for the Kerov–Kirillov–Reshetikhin bijection in the combinatorial Bethe ansatz. They also lead to general N-soliton solutions of the box–ball system.

Hirota quadratic equations for the extended Toda hierarchy

The extended Toda hierarchy (ETH) was introduced by E. Getzler [Ge] and independently by Y. Zhang [Z] in order to describe an integrable hierarchy that governs the Gromov-Witten invariants of CP1. The Lax-type presentation of the ETH was given in [CDZ]. In this article, we give a description of the ETH in terms of tau functions and Hirota quadratic equations (HQEs), also known as Hirota bilinear equations (HBEs). A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line

Variational Discretization of Euler's Elastica Problem

A discrete version of Euler's Elastica problem is formulated by a variational principle. Hirota's bilinear equations, Lax pair formalism and the exact solution are obtained explicitly. Geometrical properties are also discussed such as discrete Frenet–Serret equations.

Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra [formula omitted

In a recent paper [P. Casati, G. Ortenzi, New integrable hierarchies from vertex operator representations of polynomial Lie algebras, J. Geom. Phys. 56 (3) (2006) 418–449] Casati and Ortenzi gave a representation-theoretic interpretation of recently discovered coupled soliton equations, which were described by e.g. R. Hirota, X. Hu, X. Tang [A vector potential KdV equation and vector Ito equation: Soliton solutions, bilinear Bäcklund transformations and Lax pairs, J. Math. Anal. Appl. 288 (1) (2003) 326–348. [3]], S. Kakei [Dressing method and the coupled KP hierarchy, Phys. Lett. A 264 (6) (2000) 449–458. [6]] and S.Yu. Sakovich [A note in the Painlevé property of coupled KdV equation, arXiv:nlin.SI/0402004. [7]]. Casati and Ortenzi use vertex operators for these Lie algebras and a boson–fermion type of correspondence to get a hierarchy of coupled Hirota bilinear equations. In this paper we reformulate the Hirota bilinear description for the Lie algebra g l ∞ ( n ) and obtain a bilinear identity for matrix wave functions. From that it is straightforward to deduce the Sato–Wilson, Lax and Zakharov–Shabat equations. Using these wave functions and standard calculus with vertex operators we obtain elementary Bäcklund–Darboux transformations.

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 A GENERALIZED TZITZEICA EQUATION

Extending the so called coupled KP hierarchy to negative time flows one obtains coupled Toda-type equations defined on a two-dimensional lattice. These equations allow for reductions to 1+1 dimensional integrable systems that are defined on a finite part of this lattice. A system of coupled Hirota bilinear equations, obtained from such a reduction and defined on only 5 points of the lattice, will be shown to correspond to a coupling of a Tzitzeica equation to two linear equations. The Lax representation of this system is also presented.

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.

Complexiton solutions to integrable equations

Complexiton solutions (or complexitons for short) are exact solutions newly introduced to integrable equations are newly introduced. Starting with the solution classification for a linear differential equation, the Korteweg–de Vries equation and the Toda lattice equation are considered as examples to exhibit complexiton structures of nonlinear integrable equations. The crucial step in the solution process is to apply the Wronskian and Casoratian techniques for Hirota's bilinear equations. Correspondence between complexitons of the Korteweg–de Vries equation and complexitons of the Toda lattice equation is provided.

Toroidal Lie algebras and Bogoyavlensky's 2 + 1-dimensional equation

We introduce an extension of the ℓ-reduced KP hierarchy, which we call the ℓ-Bogoyavlensky hierarchy. Bogoyavlensky's 2 + 1-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of ℓ = 2. We present a group-theoretic characterization of this hierarchy on the basis of the 2-toroidal Lie algebra sℓℓtor⁠. This reproduces essentially the same Hirota bilinear equations as those recently introduced by Billig and Iohara et al. We can further derive these Hirota bilinear equations from a Lax formalism of the hierarchy. This Lax formalism also enables us to construct a family of special solutions that generalize the breaking soliton solutions of Bogoyavlensky. These solutions contain the N-soliton solutions, which are usually constructed by use of vertex operators.

Multisoliton solutions and integrability aspects of coupled higher-order nonlinear Schrodinger equations

Using Painleve singularity structure analysis, we show that coupled higher-order nonlinear Schrodinger (CHNLS) equations admit the Painleve property. Using the results of the Painleve analysis, we succeed in Hirota bilinearizing the CHNLS equations for the integrable cases. Solving the Hirota bilinear equations, one soliton and two soliton solutions are explicitly obtained. Lax pairs are explicitly constructed.

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.

Coloring random triangulations

We introduce and solve a two-matrix model for the tri-coloring problem of the vertices of a random triangulation. We present three different solutions: (i) by orthogonal polynomial techniques, (ii) by use of a discrete Hirota bilinear equation, (iii) by direct expansion. The model is found to lie in the universality class of pure two-dimensional quantum gravity, despite the non-polynomiality of its potential.

Analytic Bethe ansatz and functional equations for Lie superalgebra

From the point of view of the Young superdiagram method, an analytic Bethe ansatz is carried out for Lie super algebra . For the transfer matrix eigenvalue formulae in dressed-vacuum form, we present some expressions, which are quantum analogues of Jacobi - Trudi and Giambelli formulae for Lie superalgebra . We also propose transfer-matrix functional relations, which are Hirota bilinear difference equations with some constraints.