The Interaction of Shocks with Dispersive Waves II. Incompressible-Integrable Limit

This is the second in a two-part series of articles in which we analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading-order shock equations and the leading-order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading-order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading-order shock equations.     

Near-barrier fusion reactions induced by α on197Au,193Ir,191Ir,185Re,181Ta,121Sb and69Ga nuclei

Fusion-evaporation cross-sections for the α-induced reactions upon¹⁹⁷Au,¹⁹³Ir,¹⁹¹Ir,¹⁸⁵Re,¹⁸¹Ta,¹²¹Sb and⁶⁹Ga nuclei at bombarding energies near the Coulomb barrier have been measured by off-line observation of the γ-rays emitted in the radioactive decay of the residual nuclei using stacked foil technique. The total fusion cross-section for the systems have been compared with simple statistical model calculations using the code ALICE/91 as well as with the coupled channel calculations that include the β₂ and ν₄ slatic deformations and dynamic couplings of the vibrational/rotational states of the target and the projet tile using the code CCDEF.     

Fixed-frequency and threshold molecular photoelectron spectra of S 1s, Si 1s and Cl 1s core levels

Fixed-frequency and threshold photoelectron spectra have been recorded for ionization from the S ls shells in SF₆, CS₂ and COS, the Si 1s shells in SiH₄ and SiCl₄ and the Cl 1s shell in SiCl₄ using synchrotron radiation. Fixed-frequency spectra generally showed a single strong ionization feature with associated weak satellite structure due to excited ionized states. Threshold spectra closely resembled X-ray absorption spectra but with an additional feature due to direct ionization. In cases where resonant process enhanced the NEXAFS spectrum direct ionization was not observed.     

New derivatives of coumarin: 2-(N-R-imino)-2H-1-benzopyrans

Methods of synthesizing previously unknown derivatives of coumarin — 2-(N-R-imino)-2H-1-benzopyrans, where R = Alk, Ar — are proposed. Possibilities of alternative synthetic schemes are discussed, and hypotheses concerning probable mechanisms of the reactions are formulated.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 6, pp. 760–766, June, 1994. Original article submitted May 23, 1994.     

Chemical composition and reaction analysis of single aerosol particles

In situ measurements of gas-liquid surface reactions of single aerosol microdroplets are presented. By means of optical levitation in combination with elastic (Mie) and inelastic (Raman) light scattering it is possible to get information on the chemistry of e.g. acid/base reactions as well as the physical behavior of single microparticles.     

Thermal activation of complexation reaction on aerosil surface

The influence of temperature in the 20–100°C range on processes of bimolecular interaction in the aerosil-phenanthrene-trinitrobenzene system has been studied. Temperature-stimulated CTC accumulation on the surface is found. Activation energies of complex formation are estimated. Quenching of phenanthrene fluorescence by acceptor has a mixed static and dynamic character.     

Reactivity of sites on Al2O3 surface modified by Ni

It has been established that CO oxidation with excess O₂ on Ni/Al₂O₃ forms an inhomogeneous NiO contact surface that smoothes critical effects in the sharp reactivity change.     

The Assessment of Reservation Prices of an Intermediate R&D Result and Patent Reward Under Oligopolistic Competitive Behaviour

One of the shortcomings of R&D evaluation is a lack of emphasis on analytical assessment of the value of an on-going R&D project. This paper addresses the problem concerning the distributive aspect of access to superior knowledge. Decision tree analysis and probability models appear to be appropriate tools for assessing the values of an intermediate result and patent reward of a firm's R&D decision in an environment of perfect information and oligopolistic competition. The assessed values are used as minimum prices acceptable to the firm when the knowledge is disseminated to the public. This paper will attempt to resolve this problem through the determination of the appropriate values of the reservation price of the first-stage invention in terms of the final reward and of a patent reward for which the inventor is willing to apply.     

Balance constraints reduction of large-scale linear programming problems

This paper describes a method and the corresponding algorithms for simplification of large-scale linear programming models. It consists of the elimination of the balance constraints (i.e. constraints with zero RHS term). The idea is to apply some linear transformations to the original problem in order to nullify the balance constraints. These transformations are able to simultaneously eliminate more balance rows. The core of this contribution is the introduction of the reduction matrix and the associated theorems on the equivalent linear programs (original and reduced). The numerical experiments with this method of simplification proved this approach to be beneficial for a large class of LP problems.This research work was done while the first author was at Duisburg University, Mess-, Steuer und Regelungstechnik, Germany, under the greatly appreciated financial assistance given by the Alexander-von-Humboldt Foundation.     

The pole expansion in normalized QED

We present a pole expansion for the propagators in the framework of normalized quantum electrodynamics and compare it with the more canonical results from S-matrix theory.