Singular Ray Research Articles

Non-Markovian speedup evolution of a center massive particle in two-dimensional environmental model

A two-dimensional ray model is introduced to realize the non-Markovian speedup evolution of a center massive particle gravitationally coupled to a controllable environment (multilayer arrangement of the massive particles). By controlling the environment, for instance by choosing a judicious mass of the environmental particles or by changing the separation distance of each massive particle, two dynamical crossover behaviors from Markovian to non-Markovian and from no-speedup to speedup are achieved due to the gravitational interactions between the system particle and environmental particles. It is obvious that the critical mass of the environmental particles or the critical separation distance for these two dynamical crossover behaviors restrict each other directly. The larger the value of the mass of the environmental particles is, the smaller the value of the critical separation distance should be requested. In addition, it should be emphasized that the non-Markovian dynamics is the principal physical reason for the speedup evolution of the system massive particle. Particularly, the non-Markovianity of the dynamics process of the system massive particle in the even ray case has better correspondence with the quantum speed limit time than that in the singular ray case.

Steady-state ballistic thermal transport associated with transversal motions in a damped graphene lattice subjected to a point heat source

In the paper we deal with ballistic heat transport in a graphene lattice subjected to a point heat source. It is assumed that a graphene sheet is suspended under tension in a viscous gas. We use the model of a harmonic polyatomic (more exactly diatomic) lattice performing out-of-plane motions. The dynamics of the lattice is described by an infinite system of stochastic ordinary differential equations with white noise in the right-hand side, which models the point heat source. On the base of the previous analytical unsteady analysis an analytical formula in continuum approximation is suggested, which allows one to describe a steady-state kinetic temperature distribution in the graphene lattice in continuum approximation. The obtained solution is in a good agreement with numerical results obtained for the discrete system everywhere excepting a neighbourhood of six singular rays with the origin at the heat source location. The continuum solution becomes singular at these rays, unlike the discrete one, which appears to be localized in a certain sense along the rays. The factors, which cause such a directional localization and the mismatch between the continuum and discrete solutions are discussed. We expect that the suggested formula is applicable for various damped polyatomic lattices where all particles have equal masses in the case of universal for all particles external viscosity.

Optimal Resource Allocation in a Two-Sector Economy with an Integral Functional: Theoretical Analysis and Numerical Experiments

We investigate the allocation of resources in a two-sector economic model with a Cobb–Douglas production function for different depreciation rates with an integral functional on a finite time horizon. The problem is reduced to some canonical form by the scaling of phase variables and time. The extremal solution constructed by Pontryagin’s maximum principle is shown to be optimal. For a sufficiently large planning horizon, the optimal control has two or three switch points, contains one singular section, and vanishes on the final section. A transition “calibration” regime is observed between the singular section, where the system moves along a singular ray, and the final section. The maximum-principle boundary-value problem is solved in explicit form and the solution is illustrated with graphs based on numerical calculations.

Optimal Control in the Resource Allocation Problem for a Two-Sector Economy with a CES Production Function

We consider the resource allocation problem in a two-sector economy with a Constant Elasticity of Substitution (CES) production function on a given sufficiently long finite planning horizon. The performance criterion being optimized is one of the phase coordinates at the terminal time. The scalar control u satisfies the geometrical constraint u ∈ [0, 1]. The optimal control contains a singular section. Analyzing the Pontryagin maximum principle boundary-value problem we find the extremal triple and prove its optimality using a special representation of the functional increment (the theorem of sufficient conditions of optimality in terms of maximum-principle constructs). The solution is constructed with the aid of a Lambert special function y = W(x), that solves the equation ye y = x. The optimal control consists of three sections: the initial section that involves motion toward the singular ray L sng = {x 1 = x 2 > 0} , the singular section that involves motion along the singular ray, and terminal section that involves motion under zero control. When the initial state is on the singular ray, the optimal control consists of two sections — singular and terminal. The article is related to a number of the authors’ publications using other production function.

Optimal Resource Allocation in a Two-Sector Economic Model with an Integral Functional

We investigate the resource allocation problem in a two-sector economic model with a Cobb-Douglas production function with different depreciation rates. The problem is considered on a finite time horizon with an integral type functional. Optimality of the extremum solution constructed by the Pontryagin maximum principle is established. When the planning horizon is sufficiently long, the optimal control has two or three switching points, contains one singular section, and vanishes on the terminal section. A transitional “calibration” regime exists between the singular section, where the motion is along a singular ray, and the terminal section. The solution of the maximum-principle boundary-value problem is presented in explicit form, accompanied by graphs based on numerical results.

Diffraction by polyhedral angle: Field in vicinity of singular ray

If the plane acoustic wave hits a polyhedral angle, the scattered field consists of an incident wave, of its reflections from the angle’s facets, of primary edge waves excited at angle’s edges, and of a spherical wave excited at the angle’s vertex. It is supposed for simplicity that the geometry of the problem is such that there is no secondary and subsequent edge waves. There are two types of light–shadow boundaries: first for incident and reflected waves and second for edge waves. As is well known, the field in penumbras is expressed via a Fresnel integral. The incident wave’s ray passing through the vertex and its reflections from facets are singular rays belonging to the light–shadow boundaries of both types. It will be shown that the field in the vicinity of a singular ray is expressed via a Fresnel intergal and via a generalized Fresnel integral G(p,q) which was introduced in [P. S. Clemmov and T. B. A. Senior, Proc. Cambridge Philos. Soc. 9, 570–572 (1953); see also V. A. Borovikov, Uniform Stationary Phase Method, IEE Electromagnetic Series 40, London, 1994]. This result is valid for an elastic incident wave as well.

Weak diffracted shocks near singular rays

Weak diffracted shocks near singular rays

A ray method for near-shore plumes in a shallow-water flow

Far from a shoreline, the spreading of a contaminant plume in a shallow-water flow can be predicted easily and accurately by a ray-tracing method. Unfortunately, the concentration predictions become singular at a beach, where the ray paths have a caustic. In this paper a uniform approximation is derived which remains valid at a beach. It is shown how the singular ray solutions corresponding to rays incident to and transmitted from the beach can be combined to construct the uniform approximation.

Transverse Diffraction of Nonlinear Waves and Singular Rays

We derive an asymptotic partial differential equation which describes the diffraction of a weakly nonlinear high frequency wave in a direction transverse to its rays. Special cases of this equation are the Zabolotskaya–Khokhlov equation and the Kadomtsev–Petviashvilli equation. We use the equation to analyze singular rays. We also derive generalized Zabolotskaya–Khokhlov equations for sound waves.

Conical flow near singular rays

The steady flow of an ideal gas past a conical body is investigated by the method of matched asymptotic expansions, with particular emphasis on the flow near the singular ray occurring in linearized theory. The first-order problem governing the flow in this region is formulated, leading to the equation of Kuo, and an approximate solution is obtained in the case of compressive flow behind the main front. This solution is compared with the results of previous investigations with a view to assessing the applicability of the Lighthill-Whitham theories.

Internal Hypersonic Flow

Apart from method of characteristic calculations, there has been little analytical consideration of high-speed internal flows where transverse curvature and/or viscous-inviscid interaction are important. Molder [1] has carried out a numerical computation for internal conical flow using the inviscid Taylor-Macoll equations; however, a singular ray is encountered. Minassian [2] obtained an approximate inviscid solution for internal hypersonic flow; and Lee [3] has recently obtained some numerical solutions. The purpose of the present analysis is to investigate more general hypersonic internal flows with, and without, viscous interactions.