Aspects of the Noisy Burgers Equation

The noisy Burgers equation describing for example the growth of an interface subject to noise is one of the simplest model governing an intrinsically nonequilibrium problem. In one dimension this equation is analyzed by means of the Martin-Siggia-Rose technique. In a canonical formulation the morphology and scaling behavior are accessed by a principle of least action in the weak noise limit. The growth morphology is characterized by a dilute gas of nonlinear soliton modes with gapless dispersion law with exponent z=3/2 and a superposed gas of diffusive modes with a gap. The scaling exponents and a heuristic expression for the scaling function follow from a spectral representation.Comment: 23 pages,LAMUPHYS LaTeX-file (Springer), 13 figures, and 1 table, to appear in the Proceedings of the XI Max Born Symposium on "Anomalous Diffusion: From Basics to Applications", May 20-24, 1998, Ladek Zdroj, Polan

The solution of Burgers' equation for sinusoidal excitation at the upstream boundary

This paper generates an exact solution to Burgers' nonlinear diffusion equation on a convective stream with sinusoidal excitation applied at the upstream boundary, x =0. The downstream boundary, effectively at x =∞, is assumed to always be far enough ahead of the convective front at x=V s t that no disturbance is felt there. The Hopf-Cole transformation is applied in achieving the analytical solution, but only after integrating the equation and its conditions in x to avoid a nonlinearity in the transformed upstream boundary condition.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42700/1/10665_2006_Article_BF02383570.pd

Universality classes in Burgers turbulence

We establish necessary and sufficient conditions for the shock statistics to approach self-similar form in Burgers turbulence with L\'{e}vy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski's coagulation equation with additive kernel, and upon our previous characterization of the domains of attraction of self-similar solutions for this equation

Emission spectra and intrinsic optical bistability in a two-level medium

Scattering of resonant radiation in a dense two-level medium is studied theoretically with account for local field effects and renormalization of the resonance frequency. Intrinsic optical bistability is viewed as switching between different spectral patterns of fluorescent light controlled by the incident field strength. Response spectra are calculated analytically for the entire hysteresis loop of atomic excitation. The equations to describe the non-linear interaction of an atomic ensemble with light are derived from the Bogolubov-Born-Green-Kirkwood-Yvon hierarchy for reduced single particle density matrices of atoms and quantized field modes and their correlation operators. The spectral power of scattered light with separated coherent and incoherent constituents is obtained straightforwardly within the hierarchy. The formula obtained for emission spectra can be used to distinguish between possible mechanisms suggested to produce intrinsic bistability.Comment: 18 pages, 5 figure

Statistical properties of the Burgers equation with Brownian initial velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its $n$-point correlations. In the same limit, we derive the $n-$point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.Comment: 42 pages, published in J. Stat. Phy

Structure of shocks in Burgers turbulence with L\'evy noise initial data

We study the structure of the shocks for the inviscid Burgers equation in dimension 1 when the initial velocity is given by L\'evy noise, or equivalently when the initial potential is a two-sided L\'evy process $\psi_0$. When $\psi_0$ is abrupt in the sense of Vigon or has bounded variation with $\limsup_{|h| \downarrow 0} h^{-2} \psi_0(h) = \infty$, we prove that the set of points with zero velocity is regenerative, and that in the latter case this set is equal to the set of Lagrangian regular points, which is non-empty. When $\psi_0$ is abrupt we show that the shock structure is discrete. When $\psi_0$ is eroded we show that there are no rarefaction intervals.Comment: 22 page

Complete integrability of shock clustering and Burgers turbulence

We consider scalar conservation laws with convex flux and random initial data. The Hopf-Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial data is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group that is in analogy with geodesic flow on $SO(N)$; (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for $N$-dimensional approximations of the Lax equations, and yield formulas for the limit $N \to \infty$. The main observation is that the Lax equations are a $N \to \infty$ limit of a Markovian variant of the $N$-wave model. This allows us to introduce a variety of methods from the theory of integrable systems

Correlations, soliton modes, and non-Hermitian linear mode transmutation in the 1D noisy Burgers equation

Using the previously developed canonical phase space approach applied to the noisy Burgers equation in one dimension, we discuss in detail the growth morphology in terms of nonlinear soliton modes and superimposed linear modes. We moreover analyze the non-Hermitian character of the linear mode spectrum and the associated dynamical pinning and mode transmutation from diffusive to propagating behavior induced by the solitons. We discuss the anomalous diffusion of growth modes, switching and pathways, correlations in the multi-soliton sector, and in detail the correlations and scaling properties in the two-soliton sector.Comment: 50 pages, 15 figures, revtex4 fil