Asymptotically Euclidean Manifolds and Twistor Spinors

We analyse two asymmetric discrete Painlevé equations, namely d-P_II and q-P_III which are known to be discrete forms of P_III and P_VI respectively. We show that both equations are self-dual. This means that the same equation governs the evolution along the discrete independent variable and the transformations under the action of the Schlesinger transforms along the parameters of the discrete Painlevé. A bilinear formulation of the self-dual equation is given as a system of nonautonomous Hirota–Miwa equations. Received: 15 February 1997 / Accepted: 19 June 1997     

On the Discretization of the Coupled Integrable Dispersionless Equations

We study the integrable discretization of the coupled integrable dispersionless equations. Two semi-discrete version and one full-discrete version of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented.     

Solutions to Nonlocal Integrable Discrete Nonlinear Schr?dinger Equations via Reduction

Solutions to local and nonlocal integrable discrete nonlinear Schr?dinger(IDNLS) equations are studied via reduction on the bilinear form. It is shown that these solutions to IDNLS equations can be expressed in terms of the single Casorati determinant under different constraint conditions.     

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. Received: 15 May 1996 / Accepted: 25 November 1996     

Addition Formulae of Discrete KP,q-KP and Two-Component BKP Systems

In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies,we prove that these addition formulae are equivalent to these hierarchies.These studies show that the addition formula in the research of the integrable systems has good universality.     

Elliptic Solutions to Difference Non-Linear Equations and Related Many-Body Problems

We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for τ-functions. Starting from a given algebraic curve, we express the τ-function and the Baker–Akhiezer function in terms of the Riemann theta function. We show that the elliptic solutions, when the τ-function is an elliptic polynomial, form a subclass of the general algebro-geometric solutions. We construct the algebraic curves of the elliptic solutions. The evolution of zeros of the elliptic solutions is governed by the discrete time generalization of the Ruijsenaars-Schneider many body system. The zeros obey equations which have the form of nested Bethe-ansatz equations, known from integrable quantum field theories. We discuss the Lax representation and the action-angle-type variables for the many body system. We also discuss elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda equations and describe the loci of the zeros. Received: 15 May 1997 / Accepted: 7 September 1997     

Bilinearization and new multisoliton solutions for the (4+1)-dimensional Fokas equation

The (4 + 1)-dimensional Fokas equation is derived in the process of extending the integrable Kadomtsev–Petviashvili and Davey–Stewartson equations to higher-dimensional nonlinear wave equations. This equation is under investigation in this paper. Hirota’s bilinear method is, for the first time, used to solve such a higher-dimensional equation. In order to bilinearize the Fokas equation, some appropriate transformations are adopted. As a result, single-soliton solution, double-soliton solution and three-soliton solution are obtained. A new uniform formula of n-soliton solution is derived from this. It is shown that the transformations adopted in this work play a key role in converting the Fokas equation into Hirota’s bilinear form.     

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 gl(K|M)$gl(K|M)$-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.     

Casorati determinant solution for the discrete-time relativistic Toda lattice equation

The discrete-time relativistic Toda lattice (dRTL) equation is investigated by using the bilinear formalism. Bilinear equations are systematically constructed with the aid of the singularity confinement method. It is shown that the dRTL equation is decomposed into the Bäcklund transformations of the discrete-time Toda lattice equation. The N-soliton solution is explicitly constructed in the form of the Casorati determinant.     

Masterequation,H-theorem and stationary solution for coupled quantum systems

We extend the Zwanzig projector formalism to coupled systems taking into account the mutual interactions of the reduced density matrices of both systems. In the Born- and Markoff-approximation we end up with a bilinear masterequation for occupation probabilities, in contrast to the usually studied linear equations. We derive theH-theorem for this equation and show that the stationary solution is the canonical or more generally a grand canonical density matrix.     

Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy

Abstract

In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). The Hirota's bilinear equations obtained in this paper for the KPSCS are in different forms by comparing with the existing results.     

Hedgehog structures in general quark-meson Lagrangians

We investigate stationarity properties of hedgehog configurations in effective quarkmeson Lagrangians involving scalar and vector boson fields with arbitrary self-couplings. The Lagrangians are assumed to be in theSU(2)-sector and to be invariant under Lorentz-transformations, isorotations and time reversal. Restricting ourselves to the mean field solution based on valence quarks in sphericals-orbits and bilinear couplings the quark-hedgehog is shown to generate simply structured meson fields and to correspond to a stationary solution of the equations of motion. In most practical cases this is even of minimal energy.     

From crystal steps to continuum laws: Behavior near large facets in one dimension

The passage from discrete schemes for surface line defects (steps) to nonlinear macroscopic laws for crystals is studied via formal asymptotics in one space dimension. Our goal is to illustrate by explicit computations the emergence from step motion laws of continuum-scale power series expansions for the slope near the edges of large, flat surface regions (facets). We consider surface diffusion kinetics via the Burton, Cabrera and Frank (BCF) model by which adsorbed atoms diffuse on terraces and attach-detach at steps. Nearest-neighbor step interactions are included. The setting is a monotone train of N steps separating two semi-infinite facets at fixed heights. We show how boundary conditions for the continuum slope and flux, and expansions in the height variable near facets, may emerge from the algebraic structure of discrete schemes as N→∞. Our technique relies on the use of self-similar discrete slopes, conversion of discrete schemes to sum equations, and their reduction to nonlinear integral equations for the continuum-scale slope. Approximate solutions to the continuum equations near facet edges are constructed formally by direct iterations. For elastic-dipole and multipole step interactions, the continuum slope is found in agreement with a previous hypothesis of ‘local equilibrium’.     

Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. This paper studies discrete data assimilation for the Lorenz equations and the incompressible two-dimensional Navier-Stokes equations. In both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.     

Bilinear Identities and Hirota’s Bilinear Forms for the (γn, σk)-KP Hierarchy

In this paper, we discuss how to construct the bilinear identities for the wave functions of the (γ_n, σ_k)-KP hierarchy and its Hirota’s bilinear forms. First, based on the corresponding squared eigenfunction symmetry of the KP hierarchy, we prove that the wave functions of the (γ_n, σ_k)-KP hierarchy are equal to the bilinear identities given in Sec.3 by introducing N auxiliary parameters z_i, i = 1, 2,?…?, N. Next, we derived the bilinear equations for the tau-function of the (γ_n, σ_k)-KP hierarchy. Then, we obtain the bilinear equations for the taufunction of the mixed type of KP equation with self-consistent sources (KPESCS), which includes both the first and the second type of KPESCS as special cases by setting n = 2 and k = 3. Finally, using the relation between the Hirota bilinear derivatives and the usual partial derivatives, we show the procedure of translating the Hirota’s bilinear equations into the mixed type of KPESCS.     

Perturbed beta–gamma systems and complex geometry

We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta–gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a Hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associated with Hermitian metric. We show that they decompose into linear and bilinear equations and lead to the vanishing of the first Chern class of the manifold where the system is defined. We discuss the relation of these equations to the generalized Maurer–Cartan structures related to BRST operator. Finally we describe the relations of the generalized Maurer–Cartan bilinear operation and the Courant/Dorfman brackets.     

Algebraization of difference eigenvalue equations related to Uq(sl2)

A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polynoms (discrete polynoms). Such difference operators can be constructed by means of U_q(sl₂) the quantum deformation of the sl₂ algebra. The roots of the polynoms determine the spectrum and obey the Bethe ansatz equations. A particular case of difference equations for q-hypergeometric and Askey-Wilson polynoms is discussed. Applications to the problem of Bloch electrons in a magnetic field are outlined.     

Hydrodynamic Limit of Coagulation-Fragmentation Type Models of k-Nary Interacting Particles

Hydrodynamic limit of general k-nary mass exchange processes with discrete mass distribution is described by a system of kinetic equations that generalize classical Smoluchovski's coagulation equations and many other models that are intensively studied in the current mathematical and physical literature. Existence and uniqueness theorems for these equations are proved. At last, for k-nary mass exchange processes with k>2 an alternative nondeterministic measure-valued limit (diffusion approximation) is discussed.