Formulation of scattering theory for the nonlinear Schrödinger equation with boundary conditions of the finite density type in a soliton-free sector

Asymptotic canonical variables and the classical S matrix for the nonlinear Schrödinger model with nonzero boundary conditions are calculated in terms of spectral data of an auxiliary linear operator, subject to the condition that a discrete spectrum does not exist.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 60–67, 1988.     

Liouville's theorem and the method of the inverse problem

An approach to the investigation of the Zakharov-Shabat equations is developed. This approach is based on a classical theorem of Liouville and is the synthesis of finite-zone integration, the matrix Riemann problem method and the theory of isomonodromy deformations of differential equations. The effectiveness of the proposed scheme is demonstrated by developing dressing procedures for the Bullough-Dodd equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 113–125, 1984.     

The isomonodromy approach to matric models in 2D quantum gravity

We consider the double-scaling limit in the hermitian matrix model for 2D quantum gravity associated with the measure exp . We show that after the appropriate modification of the contour of integration the Cross-Migdal-Douglas-Shenker limit to the Painlevé I equation (in the generic case of the pure gravity) is valid and calculate the nonperturbative parameters of the corresponding Painlevé function. Our approach is based on the WKB-analysis of the L-A pair corresponding to the discrete string equation in the framework of the Inverse Monodromy Method. Here we extend our results, which were obtained before for the particular casesN=2,3. Our analysis complements the isomonodromy approach proposed by G. Moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself.     

The Fisher-Hartwig Formula and Entanglement Entropy

Toeplitz matrices have applications to different problems of statistical mechanics. Recently it was used for calculation of entanglement entropy in spin chains. In the paper we review these recent developments. We use the Fisher-Hartwig formula, as well as the recent results concerning the asymptotics of the block Toeplitz determinants, to calculate entanglement entropy of large block of spins in the ground state of XY spin chain.     

Temperature correlators of the impenetrable Bose gas as an integrable system

It is shown that the temperature equal-time correlators of impenetrable bosons in one space dimension are described by a classical integrable system. Partial differential equations for two-point as well as for multipoint correlators are obtained. The short-distance and low-density expansions are constructed.     

Long-distance asymptotics of temperature correlators of the impenetrable Bose gas

The inverse scattering method is applied to the integrable nonlinear system describing temperature correlators of the impenetrable bosons in one space dimension. The corresponding matrix Riemann problems are constructed for two-point as well as for multi-point correlators. Long-distance asymptotics of two-point correlators is calculated.     

Modulation instability of solutions of the nonlinear Schrödinger equation

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 2, pp. 163–172, August, 1990.     

In Memory of Sergei Yuryevich Slavyanov

Theoretical and Mathematical Physics -     

The impact of the natural sciences on archaeology

This article reviews some of the applications of physics to the solution of archaeological problems. The use of magnetic, resistivity and electromagnetic surveying techniques for the location of buried features is described. Various methods of age determination are outlined while the problems associated with radiocarbon dating of organic material and thermoluminescent dating of pottery are discussed in detail. The techniques, including petrological examination, chemical analysis and isotopic analysis, employed in the physical examination of archaeological artefacts are described. Examples of the application of these techniques in establishing the source of the raw materials used in pottery, metal and stone implements and in elucidating the techniques of manufacture of pottery and metal objects are also given.