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We continue the study of the B-Toda hierarchy (the Toda lattice with the constraint of type B), which can be regarded as a discretization of the BKP hierarchy. We introduce the tau function of the B-Toda hierarchy and obtain bilinear equations for it. Examples of soliton tau functions are presented in explicit form. [ABSTRACT FROM AUTHOR]

We define the Dyson diffusion process on a curved smooth closed contour in the plane and derive the Fokker–Planck equation for the probability density. Its stationary solution is shown to be the Boltzmann weight for the logarithmic gas confined on the contour. [ABSTRACT FROM AUTHOR]

We study elliptic families of solutions of the recently introduced constrained Toda hierarchy, i.e., solutions that are elliptic functions of some linear combination of the hierarchical times. Equations of motion for poles of such solutions are obtained. [ABSTRACT FROM AUTHOR]

We consider solutions of the 2D Toda lattice hierarchy that are elliptic functions of the "zeroth" time . It is known that their poles as functions of move as particles of the elliptic Ruijsenaars–Schneider model. The goal of this paper is to extend this correspondence to the level of hierarchies. We show that the Hamiltonians that govern the dynamics of poles with respect to the th hierarchical times and of the 2D Toda lattice hierarchy are obtained from the expansion of the spectral curve for the Lax matrix of the Ruijsenaars–Schneider model at the marked points. [ABSTRACT FROM AUTHOR]

This is a short review of the Kadomtsev–Petviashvili hierarchies of types and . The main objects are the operator, the wave operator, the auxiliary linear problems for the wave function, the bilinear identity for the wave function, and the tau function. All of them are discussed in the paper. Connections with the usual (type- ) Kadomtsev–Petviashvili hierarchy are clarified. Examples of soliton solutions and the dispersionless limit of the hierarchies are also considered. [ABSTRACT FROM AUTHOR]

We study elliptic solutions of the semidiscrete B-version of the Kadomtsev–Petviashvili equation and derive the equations of motion of their poles. The auxiliary linear problems for the wave function are the main technical tool. [ABSTRACT FROM AUTHOR]

We consider solutions of the matrix Kadomtsev-Petviashvili (KP) hierarchy that are trigonometric functions of the first hierarchical time t1 = x and establish the correspondence with the spin generalization of the trigonometric Calogero-Moser system at the level of hierarchies. Namely, the evolution of poles xi and matrix residues at the poles aiαbiβ of the solutions with respect to the kth hierarchical time of the matrix KP hierarchy is shown to be given by the Hamiltonian flow with the Hamiltonian which is a linear combination of the first k higher Hamiltonians of the spin trigonometric Calogero-Moser system with coordinates xi and with spin degrees of freedom αiα and biβ. By considering the evolution of poles according to the discrete time matrix KP hierarchy, we also introduce the integrable discrete time version of the trigonometric spin Calogero-Moser system. [ABSTRACT FROM AUTHOR]

Using the bilinear formalism, we consider multicomponent and matrix Kadomtsev-Petviashvili hierarchies. The main tool is the bilinear identity for the tau function realized as the vacuum expectation value of a Clifford group element composed of multicomponent fermionic operators. We also construct the Baker-Akhiezer functions and obtain auxiliary linear equations that they satisfy. [ABSTRACT FROM AUTHOR]

Abstract: A criterion for estimating the homogeneity of materials for refractive X-ray optics from small-angle X-ray scattering data is proposed. A material of new type—nanoberyllium—is considered. A number of samples of beryllium materials, including commercial products, have been analyzed. The X-ray homogeneity of the materials is determined, as well as the influence of technological processes. [ABSTRACT FROM AUTHOR]

Local M operators for the classical sine-Gordon model in discrete space–time are constructed by convolution of the quantum trigonometric 4×4 R matrix with certain vectors in its “quantum” space. Components of the vectors are τ functions of the model. This construction generalizes the known representation of continuous-time M operators in terms of the classical r matrix. © 1997 American Institute of Physics. [ABSTRACT FROM AUTHOR]