Burgers-Hopf Equation Research Articles

Subgrid-scale closure for the inviscid Burgers-Hopf equation

Abstract. A method is presented for constructing effective stochastic models for the timeevolution of spatial averages in finite-difference discretizations of partial differential equations. This method relies on the existence of a time-scale separation in the dynamics of the spatial averages and fine-grid variables. The spatial averages, thus, are treated as the slow variables in the system and a stochastic mode reduction strategy can be applied to derive a closed form effective stochastic model. A conservative discretization of the Burgers-Hopf equation is used as particular example to illustrate the approach. An advantage over heretofore applications of stochastic mode reduction to spectrally discretized models is that the resulting closure is local and thus remains applicable even if the number of slow variables is large.

Large-NcConfinement and Turbulence

We suggest that the transition that occurs at large N_{c} in the eigenvalue distribution of a Wilson loop may have a turbulent origin. We arrived at this conclusion by studying the complex-valued inviscid Burgers-Hopf equation that corresponds to the Makeenko-Migdal loop equation, and we demonstrate the appearance of a shock in the spectral flow of the Wilson loop eigenvalues. This picture supplements that of the Durhuus-Olesen transition with a particular realization of disorder. The critical behavior at the formation of the shock allows us to infer exponents that have been measured recently in lattice simulations by Narayanan and Neuberger in d=2 and d=3. Our analysis leads us to speculate that the universal behavior observed in these lattice simulations might be a generic feature of confinement, also in d=4 Yang-Mills theory.

Periodic Solutions of the Burgers-Hopf Equation with Small Parameter and its Cellular Neural Network Model

This paper deals with the periodic solutions of the Cauchy problem to the Burgers-Hopf equation containing small viscosity term. We propose at first an explicit formula for the exact solution of the Cauchy problem and then we give an approximate formula. We use CNN approach for constructing the approximate solution and we find precise estimates of the remainder term.

Quasilinear Burgers-Hopf equation and Stasheff polytopes

The generating function for the numbers of faces of the nth Stasheff polytope is shown to satisfy the quasilinear Burgers-Hopf equation. Applications of this result are given.

Light Propagation in a Cole-Cole Nonlinear Medium via the Burgers-Hopf Equation

A new model of light propagation through a so-called weakly three-dimensional Cole-Cole nonlinear medium with short-range nonlocality was recently proposed. In particular, it was shown that in the geometric optics limit, the model is integrable and is governed by the dispersionless Veselov-Novikov (dVN) equation. The Burgers-Hopf equation can be obtained as a (1+1)-dimensional reduction of the dVN equation. We discuss its properties in the specific context of nonlinear geometric optics and consider an illustrative explicit example.

Discrete Approximations with Additional Conserved Quantities: Deterministic and Statistical Behavior

Discrete numerical approximations with additional conserved quantities are devel- oped here both for barotropic geophysical flows generalizing the 2D incompressible fluid equations and suitable discretizations of the Burgers-Hopf equation. Mathematical, numerical, and statistical properties of these approximations are studied below in various different contexts through the sym- biotic interaction of mathematical theory and scientific computing. The new contributions include an explicit concrete discussion of the sine-bracket spectral truncation with many conserved quanti- ties for 2D incompressible flow, a theoretical and numerical comparison with the standard spectral truncation, and a rigorous proof of convergence to suitable weak solutions in the limit as the number of Fourier modes increases. Systematic discretizations of the Burgers-Hopf equation are developed, which conserve linear momentum and a non-linear energy; careful numerical experiments regarding the statistical behavior of these models indicate that they are ergodic and strongly mixing but do not have equipartition of energy in Fourier space. Furthermore, the probability distribution of the values at a single grid point can be highly non-Gaussian with two strong isolated peaks in this distribution. This contrasts with earlier results for statistical behavior of difference schemes which conserve a quadratic energy. The issues of statistically relevant conserved quantities are introduced through a new case study for the Galerkin-truncated Burgers-Hopf model.

Statistical Mechanics for Truncations of the Burgers-Hopf Equation: A Model for Intrinsic Stochastic Behavior with Scaling

In this paper we consider both analytically and numerically several finite-dimensional approximations for the inviscid Burgers-Hopf equation. Fourier Galerkin truncation is introduced and studied as a simple one-dimensional model with intrinsic chaos and a well-defined mathematical structure allowing for an equilibrium statistical mechan- ics formalism. A simple scaling theory for correlations is developed that is supported strongly by the numerical evidence. Several semi-discrete difference schemes with similar mathematical properties conserving dis- crete momentum and energy are also considered. The mathematical properties of the difference schemes are analyzed and the behavior of the difference schemes is compared and contrasted with the Fourier Galerkin truncation. Numerical simulations are presented which show similarities and subtle differences between different finite-dimensional approximations both in the deterministic and stochastic regimes with many degrees of freedom.

A conservative box-scheme for the Euler equations

The work presented in this paper shows that the mixed-type scheme of Murman and Cole, originally developed for a scalar equation, can be extended to systems of conservation laws. A characteristic scheme for the equations of gas dynamics is introduced that has a close connection to a four operator scheme for the Burgers-Hopf equation. The results indicate that the scheme performs well on the classical test cases. The scheme has no tuning parameters and can be interpreted as the projection of an L∞-stable scheme. At steady state second order accuracy is obtained as a by-product of the box-scheme feature

Integrable particle systems in an external field and singular solutions of the inhomogeneous Burgers-Hopf equation

A connection is found between the motion of classical integrable particle systems in a external field under certain initial conditions and solutions to the inhomogeneous Burgers-Hopf equation, which have pole singularities. This connection allows us in some new cases to obtain solutions to the equations of motion for systems with an arbitrary number of particles.