Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schrödinger problem and the KPI equation

The resolvent operator of the linear problem is determined as the full Green function continued in the complex domain in two variables. An analog of the known Hilbert identity is derived. We demonstrate the role of this identity in the study of two-dimensional scattering. Considering the nonstationary Schrödinger equation as an example, we show that all types of solutions of the linear problems, as well as spectral data known in the literature, are given as specific values of this unique function — the resolvent function. A new form of the inverse problem is formulated.This is the last work in which M. C. Polivanov participated. In loving memory. M. B., F. P., A. P.Dipartimento di Fisica dell'Università e Sezione INFN, 73100 Lecce, ITALIA. Steklov Mathematical Institute, Vavilov Str. 42, Moscow 117966, GSP-1, Russia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 2, pp. 181–210, November, 1992.     

Singular solutions of the KdV equation and the inverse scattering method

The paper is devoted to the construction of singular solutions of the KdV equation. The presentation is based on a variant of the inverse scattering method for singular solutions of nonlinear equations developed in previous works of the authors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 133, pp. 17–37, 1984.In conclusion we wish to express our gratitude to F. Calogero, who drew our attention to works [4–6], an to mention that the generalization of these works proposed here is based on the early paper [13] of L. D. Faddeev.     

Turbulization of the wake behind a single roughness element on a blunted body at a hypersonic Mach number

The influence of a cylinder-shaped single roughness element on the laminar–turbulent transition in the presence of an entropy layer is experimentally studied. The experiments are performed on a blunted cone model at the Mach number M = 5. The roughness element is located on the blunted tip of the model. Information about the mean and fluctuating parameters of the boundary layer in the wake behind the roughness element is obtained by using hot-wire anemometry. It is shown that flow turbulization behind the roughness elements occurs at the local Reynolds number calculated on the basis of the roughness element height and equal to 400–500. It is found that the presence of the roughness element exerts a significant effect on the unsteady characteristics of the boundary layer if the roughness element height is smaller than the effective value.     

Calculating pressure fields on the basis of PIV-measurements for supersonic flows

The velocity fields obtained by PIV (Particle Image Velocimetry) in supersonic flows are not sufficient to determine the integral characteristics of the flow. Additional data, for example, on pressure can be obtained from the solution of the Navier?Stokes equations. For incompressible flows, the solution of these equations is not too complicated. However, for supersonic flows, the need to take into account the flow density and the increasing number of experimental errors make it more difficult. This paper proposes a new method for calculating density and pressure from PIV data on the basis of the continuity equation. This method is robust and easy to implement for compressible flows.     

Application of “POLIS” PIV system for measurement of velocity fields in a supersonic flow of the wind tunnels

Measurement results on the mean velocity fields and fields of velocity pulsations in the supersonic flows obtained by means of the PIV measurement set “POLIS” are presented. Experiments were carried out in the supersonic blow-down and stationary wind tunnels at the Mach numbers of 4.85 and 6. The method of flow velocity estimate in the test section of the blow-down wind tunnel was grounded by direct measurements of stagnation pressure in the setup settling chamber. The size of tracer particles introduced into the supersonic flow by a mist generator was determined; data on the structure of pulsating velocity in a track of an oblique-cut gas-dynamic whistle were obtained under the conditions of self-oscillations.