Hopf Equation Research Articles

A Probabilistic Look at the Wiener--Hopf Equation

Existence, uniqueness, and asymptotic properties of solutions Z to the Wiener--Hopf integral equation $Z(x)$ $ =$ $z(x)+\int_{-\infty}^xZ(x-y)F(dy)$, $x\ge 0$, are discussed by purely probabilistic methods, involving random walks, supermartingales, coupling, the Hewitt--Savage 0--1 law, ladder heights, and exponential change of measure.

Set–Valued Implicit Wiener-Hopf Equations and Generalized Strongly Nonlinear Quasivariational Inequalities

In this paper, we study a class of generalized strongly nonlinear quasivariational inequality problem (GSNQV IP (T,A, g,D,K(x))) which include the most of quasivariational inequalities and quasicomplementarity problems as special cases. We prove that the generalized strongly nonlinear quasivariational inequality problem is equivalent to solving the set-valued implicit Wiener–Hopf equation. By using the equivalence, a new iterative algorithm for finding the approximate solutions of the generalized strongly nonlinear quasivariational inequality problems are suggested and analyzed. The convergence criteria for the algorithm is also discussed. These new results include many known results for generalized quasivariational inequalities and generalized quasicomplementarity problems as special cases. 1991 Mathematics Subject Classification. 49A29, 65J15, 47H17.

Diffraction of acoustic waves by a semi-infinite cylindrical impedance pipe of certain wall thickness

An asymptotic high-frequency solution is presented for the problem of diffraction of acoustic waves emanating from a ring source by a semi-infinite cylindrical pipe of certain wall thickness having different internal, external and end surface impedances. Through the application of the Fourier-transform technique in conjunction with the Mode Matching method, the diffraction problem is described by a modified Wiener–Hopf equation of the second kind and then solved approximately. Various numerical results illustrating the effects of the parameters of the problem on the diffraction phenomenon are presented.

The Small Dispersion Limit for a Nonlinear Semidiscrete System of Equations

Numerical experiments that illustrate the emergence of oscillatory behavior in the solutions of a dispersive scheme approximating the Hopf equation are presented. The oscillations arise at the same time that the classical solution of the Hopf equation develops a singularity and generally have a spatial period equal to twice the grid size. Modulation equations for these period‐two solutions are derived. The modulation equations have both a hyperbolic and an elliptic region. The period‐two oscillations break down after they enter the elliptic region, and the solution blows up. We give a local description of the blowup by an exact solution. This kind of phenomenon (the blowup) has not been observed for integrable schemes. The modulation equations also have the unusual feature that they admit (some) shocks when crossings of characteristics in the hyperbolic regime occur. Other crossings lead to breakdown of the binary oscillation description, with oscillatory behavior of a more complicated nature arising.

Preconditioners for Wiener--Hopf Equations with High-Order Quadrature Rules

We consider solving the Wiener--Hopf equations with high-order quadrature rules by preconditioned conjugate gradient (PCG) methods. We propose using convolution operators as preconditioners for these equations. We will show that with the proper choice of kernel functions for the preconditioners, the resulting preconditioned equations will have clustered spectra and therefore can be solved by the PCG method with superlinear convergence rate. Moreover, the discretization of these equations by high-order quadrature rules leads to matrix systems that involve only Toeplitz or diagonal matrix--vector multiplications and hence can be computed efficiently by FFTs. Numerical results are given to illustrate the fast convergence of the method and the improvement on accuracy by using higher-order quadrature rule. We also compare the performance of our preconditioners with the circulant integral operators.

Rawlins' problem for half-plane diffraction: its generalized eigenfunctions with real wave numbers

This paper deals with a famous diffraction problem for a single half-plane Σ: x>0, y=0 as an obstacle and for some time-harmonic plane incident wave field. Rawlins in 1975 was the first to solve the mixed (Dirichlet/Neumann) boundary value problem for the scalar Helmholtz equation. He also was the first to solve the equivalent pair of coupled Wiener–Hopf equations explicitly by factoring their discontinuous 2×2 Fourier matrix symbol in 1980. Although for real wave numbers k the usual factorization procedure fails it will serve as the basis: Following the lines given by Ali Mehmeti in his habilitation thesis [1] for the (Dirichlet/Dirichlet) boundary value problem we combine the idea of integral path deforming along the branch cuts of the characteristic square root √(ξ2−k2) given in Meister's book [13] with the modern Wiener–Hopf method solution derived by Speck [24] explicitly in a H1+ε, ε⩾0, Sobolev space setting. The symmetry of the intermediate spaces Hs, H-s, ∣s∣<1 2, which is due to generalized factorization, plays a key role in deforming the Fourier integral paths in order to get Laplace transform representations of the generalized eigenfunctions of the problem. As a remarkable fact 0<ε<¼ must hold here. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Finding a Zero of The Sum of Two Maximal Monotone Operators

In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brezis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.

Two waveguide trifurcation problems

We consider the diffraction of the dominant acoustic wave mode which propagates out of the mouth of a semi-infinite waveguide made of a soft and hard half plane. This semi-infinite waveguide is symmetrically located inside an infinite waveguide whose infinite plates are soft and hard. The whole system constitutes a trifurcated waveguide. Another trifurcated waveguide is obtained by interchanging the infinite plates. A closed form solution of the resulting matrix Wiener–Hopf equation is obtained for each configuration. Thus we present exact closed-form solutions to two new waveguide trifurcation problems.

Stochastic analysis method for hopf's equation ut+ uux=0

It is proved that the stochastic charactcrtstics of Burgers’ equation u t+uux=μuxxconverge in probability to the character is tcs of Hopf ’ s cqua lion ut+uux=0 as the viscosity μ →0. It follows naturally that the solution of Burgers’ equation converges to the solution of Hopf 's equation satisfying entropy condition. This is well known result due to E. Hopf in 1950. The method here is new. This paper suggests that this method may be useful for proving the validity of “vanishing viscosity method” in certain cases. Those problems are usually extremely difficult.

Partially stiffened elastic half-plane with an edge crack

In the present paper two contact problems referring to the partially stiffened elastic half-plane along its edge with a reinforcement (or stringer) either in the form of a rigid plate or a smooth rigid stamp, are considered. The half-plane contains an edge crack which is arbitrarily pressurized, and is perpendicular to the bounding edge of the half-space. It is assumed that the mid-point of the stringer is located in the axis of the crack. Each of the above two half-plane contact problems is first reduced to a system of two singular integral equations with fixed singularities. Then by employing the generalized method of integral transforms, this system is further reduced to a system of Wiener–Hopf equations that is equivalent to the Riemann matrix boundary value problem. Exact analytical solutions of the two problems are presented in series form. Asymptotic approximations for the stress intensity factor and the energy release rate at the crack tip are also given. Finally, numerical results for the contact stresses, crack opening displacements, stress intensity factor and crack energy are displayed.

Large oscillations arising in a dispersive numerical scheme

We study the oscillatory behavior that arises in solutions of a dispersive numerical scheme for the Hopf equation whenever the classical solution of the equation develops a singularity. Modulation equations are derived that describe period-two oscillations so long as the solution of those equations takes values for which the equations are hyperbolic. However, those equations have an elliptic region that may be entered by its solutions in a finite time, after which the corresponding period-two oscillations are seen to break down. This kind of phenomenon has not been observed for integrable schemes. The generation and propagation of period-two oscillations are asymptotically analyzed and a matching formula is found for the transition between oscillatory and nonoscillatory regions. Modulation equations are also presented for period-three oscillations. Numerical experiments are carried out that illustrate our analysis.

A Canonical Diffraction Problem With Two Media

A Wiener–Hopf equation in L2 being equivalent [5] to a boundary value problem (of the first kind) for a wave-scattering Sommerfeld half-plane Σ=ℝ+×{0} which faces two different media Ω-: x2<0, Ω+: x2>0, as a special configuration in [3], is solved by canonical Weiner–Hopf factorization of its L2-regular scalar symbol γo=γo- γo+. The factors are calculated by solving a Riemann–Hilbert boundary value problem on the semi-infinite branch cuts of tj(ξ):=(ξ2−k2j)1/2, kj∈ℂ++ for j=1,2: taken parallel to the imaginary axis. The procedure following this idea is known as the Wiener–Hopf–Hilbert(–Hurd) method [2] and requires the evaluation of elliptic-type integrals. Formula (3.7) seems not to be contained in tables of integrals.

Symmetry and Scaling of Turbulent Mixing.

The stationary condition (Hopf equation) for the $(n+1)$-point correlation function of a passive scalar advected by turbulent flow is argued to have an approximate $\mathrm{SL}\left(n,R\right)$ symmetry which provides a starting point for our phenomenological theory in which less symmetric terms are treated perturbatively. The large scale anisotropy is found to be a relevant field, in contradiction with Kolmogorov phenomenology, but in agreement with the large scalar skewness observed in shear flows. Exponents are not universal, yet quantitative predictions for experiments to test the $\mathrm{SL}\left(n,R\right)$ symmetry can be formulated in terms of the correlation functions.

The master field in generalised QCD 2

As an illustration of the formalism of the master field we consider generalised QCD 2. We show how Wilson loop averages for an arbitrary contour can be computed explicitly and with some ease. A generalised Hopf equation is shown to govern the behaviour of the eigenvalue density of Wilson loops. The collective field description of the theory is therefore deduced. Finally, the non-trivial master gauge field and field strengths are obtained. These results do not seem easily accessible with conventional means.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

PERTURBING CONFORMAL TURBULENCE

We consider perturbations of the nonunitary minimal model solutions of two-dimensional conformal turbulence proposed by Polyakov. Demanding the absence of nonintegrable singularities in the resulting theories leads to constraints on the dimension of the perturbing operator. We give some general solutions of these constraints, illustrating with examples of specific models. We also examine the effect of such perturbations on the Hopf equation and derive the interesting result that the latter is invariant under a certain class of perturbations, to first-order in perturbation theory, examples of which are given in specific cases.

The radiation of sound from a finite ring-forced cylindrical elastic shell I. Wiener–Hopf analysis

The radiation of sound from a finite fluid-loaded cylindrical elastic shell is examined by means of the modified Wiener–Hopf technique. The shell is freely submerged in an inviscid compressible stationary fluid of infinite extent and axisymmetrically excited by a ring force. The problem is formulated as a three-part Wiener–Hopf equation with three unknown functions, and is recast as two uncoupled Fredholm integral equations of the second kind by utilizing a factorization and decomposition procedure. When the product of the cylinder length l and the wave number k is large, asymptotic analysis can be applied to the integral equations, which reduces them to a finite system of algebraic equations corresponding to a set of unknown constants, including the edge displacements. The method provides a straightforward formulation for the solution and is valid for arbitrary fluid loading. Numerical results are presented for the radiated far-field velocity potential. This requires a numerical decomposition of the Wiener–Hopf kernel and the solution of the set of simultaneous equations.

Burgers turbulence with random forcing: Similarity functional solution of the Hopf equation.

For the problem of Burgers turbulence with random forcing, a similarity functional solution of the Hopf equation is presented and compared with scaling arguments and replica Bethe-ansatz treatments. The corresponding field theory is almost nonanomalous. In one dimension the local fluctuations develop self-similar time-dependent behavior, while relative fluctuations within the correlation length form a steady state with Gaussian distribution. This is the precise meaning of the so-called fluctuation-dissipation theorem. The one-dimensional properties are also studied numerically. It is shown that the fluctuation-dissipation theorem is invalid above one dimension and higher-order cumulants are nonzero. In two dimensions the cumulants exhibit a logarithmic spatial dependence, which is close to but different from that in the Edwards-Wilkinson case. No other similarity functional solution is found, which may indicate that the ``strong-coupling'' results are not described by the forced Burgers equation.

Large N gauge theory — Expansions and transitions

We use solvable two-dimensional gauge theories to illustrate the issues in relating large N gauge theory to string theory. We also give an introduction to recent mathematical work which allows constructing master fields for higher dimensional large N theories. We illustrate this with a new derivation of the Hopf equation governing the evolution of the spectral density in matrix quantum mechanics. Based on lectures given at the 1994 Trieste Spring School on String Theory, Gauge Theory and Quantum Gravity.