Backscattering in the vicinity of a mode cutoff

In a recent paper to this journal (Whitman A M et al 2003 Waves Random Media 13 269-86) we derived a set of coupled equations that describe the intermodal scattering of acoustic radiation in a duct whose speed of sound varies randomly in space and time. In the paper we were mainly interested in modes that were not near cutoff. Here we study the solution of these equations in the vicinity of the cutoff. We find that near cutoff almost all the energy is reflected back independent of the other duct parameters. In addition to presenting these results, we analyse the mathematical structure of the equations in these regions in order to elucidate the reason for this behaviour.     

The derivation and solution of the modal flux equations for backscattering in a duct in the presence of random space and time fluctuations

Abstract

In this paper we first derive the equations governing the energy fluxes propagating in each of the modes of a duct. In each mode there is a forward and backward component and the equations are intended to treat ducts in which backscattering plays a major role. The modal fluxes are coupled since there is transfer of energy between the modes that occurs as a result of random time and space sound-speed fluctuations in the medium in the duct. Since the fluctuations are both space and time dependent the governing equations are radiation transport equations. This is not the case if the fluctuations depend only on space. The basic method is to develop a coupled set of equations for the energy spectra in the modes and then to integrate over the frequency to obtain the fluxes. In the second section of this paper the modal flux equations are solved. A numerical result is presented to show how energy is transferred between modes. It is also shown how the reflected energy varies as a function of duct length.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

Universal Bounds for the Littlewood-Paley First-Order Moments of the 3D Navier-Stokes Equations

We derive upper bounds for the infinite-time and space average of the L ¹-norm of the Littlewood-Paley decomposition of weak solutions of the 3D periodic Navier-Stokes equations. The result suggests that the Kolmogorov characteristic velocity scaling, U_k ~ e^1/3 k^-1/3{\mathbf{U}_\kappa\sim\epsilon^{1/3} \kappa^{-1/3}} , holds as an upper bound for a region of wavenumbers near the dissipative cutoff.     

Two new integrable couplings of the soliton hierarchies with self-consistent sources

A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with sl(4) has been presented (Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613). Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl(4). In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.     

Contiguity relations for discrete and ultradiscrete Painlevé equations

Abstract

We show that the solutions of ultradiscrete Painlevé equations satisfy contiguity relations just as their continuous and discrete counterparts. Our starting point are the relations for q-discrete Painlevé equations which we then proceed to ultradiscretise. In this paper we obtain results for the one-parameter q-P_III, the symmetric q-P_IV and the q-P_IV. These results show that there exists a perfect parallel between the properties of continuous, discrete and ultradiscrete Painlevé equations.     

Integrability and Explicit Solutions in Some Bianchi Cosmological Dynamical Systems

Abstract

The Einstein field equations for several cosmological models reduce to polynomial systems of ordinary differential equations. In this paper we shall concentrate our attention to the spatially homogeneous diagonal G ₂ cosmologies. By using Darboux’s theory in order to study ordinary differential equations in the complex projective plane ??² we solve the Bianchi V models totally. Moreover, we carry out a study of Bianchi VI models and first integrals are given in particular cases.     

AVERAGING IN WEAKLY COUPLED DISCRETE DYNAMICAL SYSTEMS

In [Y. Kifer, Averaging in difference equations driven by dynamical systems, Asterisque 287 (2003) 103–123] a general averaging principle for slow-fast discrete dynamical systems was presented. In this paper we extend this method to weakly coupled slow-fast systems. For this setting we obtain sharper estimates than in the mentioned paper.     

Solutions of Eliashberg equations for an electron-phonon coupling with a cutoff

In this paper we discuss the Eliashberg equations for the case of an electron-phonon coupling with an energy cutoff. This cutoff is imposed either for the energy difference by means of a strip function, or for both energies, with a Cooper-like expression. The strip function cutoff requires explicit calculation of not only the frequency renormalization functionZ but also the energy renormalizationX. The physical origin of such cutoffs might lie in the very strong electron-electron interaction which seems typical for highT _c superconductivity. If such cutoffs are admitted, the hypothesis thatT _c is caused at least in part by a strong electron-phonon interaction can be reconsidered. We find that the combination of strong coupling and low-energy cutoff could produce highT _c with only small isotope effect and with little damping or pulling of the phonon modes. Correlation with other physical properties, such as specific heat, is reexamined in view to estimate the coupling constant . Some objections to the model using strong electron phonon interaction are removed and better agreement with observed properties is obtained     

Triangular Newton Equations with Maximal Number of Integrals of Motion

Abstract

We study two-dimensional triangular systems of Newton equations (acceleration = velocity-independent force) admitting three functionally independent quadratic integrals of motion. The main idea is to exploit the fact that the first component M ₁(q ₁) of a triangular force depends on one variable only. By using the existence of extra integrals of motion we reduce the problem to solving a simultaneous system of three linear ordinary differential equations with nonconstant coefficients for M ₁(q ₁). With the help of computer algebra we have found and solved these ordinary differential equations in all cases. A complete list of superintegrable triangular equations in two dimensions is been given. Most of these equations were not known before.     

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R ^d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models.     

Solvable and/or Integrable and/or Linearizable N-Body Problems in Ordinary (Three-Dimensional) Space. I

Abstract

Several N -body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (“acceleration equal force;” in most cases, the forces are velocity-dependent) and are amenable to exact treatment (“solvable” and/or “integrable” and/or “linearizable”). These equations of motion are always rotation-invariant, and sometimes translation-invariant as well. In many cases they are Hamiltonian, but the discussion of this aspect is postponed to a subsequent paper. We consider “few-body problems” (with, say, N =1,2,3,4,6,8,12,16,...) as well as “many-body problems” (N an arbitrary positive integer). The main focus of this paper is on various techniques to uncover such N -body problems. We do not discuss the detailed behavior of the solutions of all these problems, but we do identify several models whose motions are completely periodic or multiply periodic, and we exhibit in rather explicit form the solutions in some cases.     

On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions

In this paper, we are interested in the influence of grazing collisions, with deflection angle near π/2, in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-sth-power force laws, and we deal with general initial data with bounded mass, energy, and entropy. First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, fors ≥ 7/3. Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value π/2. For very soft interactions, 2<s<7/3, the existence of weak solutions is discussed concerning the Boltzmann equation perturbed by a diffusion term     

Correctors for the Homogenization of Monotone Parabolic Operators

Abstract

In the homogenization of monotone parabolic partial differential equations with oscillations in both the space and time variables the gradients converges only weakly in L ^p. In the present paper we construct a family of correctors, such that, up to a remainder which converges to zero strongly in L ^p, we obtain strong convergence of the gradients in L ^p.     

Ultrahigh-energy cosmic rays, superheavy long-lived particles, and matter creation after inflation

Cosmic rays of the highest energy, above the Greisen-Zatsepin-Kuzmin (GZK) cutoff of the spectrum, may originate in decays of superheavy long-lived particles. We conjecture that these particles may be produced naturally in the early Universe from vacuum fluctuations during inflation and may constitute a considerable fraction of cold dark matter. We predict a new cutoff in the ultrahigh-energy cosmic ray spectrum E _cutoff<m _inflaton≈10¹³ GeV, the exact position of the cutoff and the shape of the cosmic ray spectrum beyond the GZK cutoff being determined by the QCD quark/gluon fragmentation. The Pierre Auger Project installation may in principle observe this phenomenon. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 4, 255–259 (25 August 1998)     

Blow-up,Zero <Emphasis Type="Italic">α</Emphasis> Limit and the Liouville Type Theorem for the Euler-Poincaré Equations

In this paper we study the Euler-Poincaré equations in . We prove local existence of weak solutions in , and local existence of unique classical solutions in , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation (α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution is u=0; for α= 0 any weak solution is u=0.     

Design and Analysis of InGaN-GaN Modulation Doped Field-Effect Transistors (MODFETs) for Over 60 GHz Operation

A Modulation-Doped Field-Effect Transistor (MODFET) structure realized in InGaN-GaN material system is presented for the first time. An analytical model predicting the transport characteristics of the proposed MODFET structure is given in detail. Electron energy levels inside and outside the quantum well channel of the MODFET are evaluated. The two-dimensional electron gas (2DEG) density in the channel is calculated by self-consistently solving Schrödinger and Poisson's equations simultaneously. Analytical results of the current-voltage and transconductance characteristics are presented. The unity-current gain cutoff frequency (f _T) of the proposed device is computed as a function of the gate voltage V _G . The results are compared well with experimental f _T value of a GaN/AlGaN HFET device. By scaling the gate length down to 0.25 m the proposed InGaN-GaN MODFET can be operated up to about 80GHz. It is shown in this paper that InGaN-GaN system has small degradation in f _T as the operating temperature is increased from 300°K to 400°K.     

Formulation of rough-surface scattering theory in terms of phase factors and approximate solutions based on this formulation

Abstract

This paper presents the formulation of rough-surface scattering theory in which the bounded phase shift factors, ζ(r, α) ζ exp[iαζ(r)], replace the elevation, ζ(r). Both the Dirichlet and the Neumann problems are considered. The integral equations for secondary surface sources are obtained that contain only this phase function in their kernels.

The Neumann (iterative) series for the solutions of the integral equations thus derived are functional Taylor series in powers of L(r, α), not in powers of ζ. If we expand L(r, α) in these series in powers of ζ(r), we obtain the standard perturbation theory series. Thus, the new formulation corresponds to the partial summation of the perturbation series.

Using the Neumann series, we obtain several uniform (with respect to αζ) approximate solutions that contain, as limiting cases, Bragg scattering, the Kirchhoff approximation, and most known advanced approximations.

In the case of random surface z = ζ(r), these new expansions contain the function ζ(r) only in the exponents, and, therefore, the result of averaging can be expressed only in terms of the characteristic functions of the multivariate probability distribution of elevations.     

Incompressible Flows of an Ideal Fluid¶with Unbounded Vorticity

In this paper we study solutions to the Euler equations of an ideal incompressible fluid in R ⁿ singular at the origin with a finite symmetry group. For an “admissible” class of finite groups we prove a local existence and uniqueness theorem. In even dimensions this theorem covers some symmetric flows with essentially unbounded vorticity. In arbitrary dimension (including n=3) we construct local in time solutions with vorticity that behaves, e.g., like a function of homogeneous degree zero near the origin. The symmetry condition provides necessary additional cancellations and is preserved by the evolution due to uniqueness. Received: 31 March 1999 / Accepted: 10 July 2000