18 research outputs found
Velocity fluctuations in forced Burgers turbulence
We propose a simple method to compute the velocity difference statistics in
forced Burgers turbulence in any dimension. Within a reasonnable assumption
concerning the nucleation and coalescence of shocks, we find in particular that
the `left' tail of the distribution decays as an inverse square power, which is
compatible with numerical data. Our results are compared to those of various
recent approaches: instantons, operator product expansion, replicas.Comment: 10 pages latex, one postcript figur
Solitons and diffusive modes in the noiseless Burgers equation: Stability analysis
The noiseless Burgers equation in one spatial dimension is analyzed from the
point of view of a diffusive evolution equation in terms of nonlinear soliton
modes and linear diffusive modes. The transient evolution of the profile is
interpreted as a gas of right hand solitons connected by ramp solutions with
superposed linear diffusive modes. This picture is supported by a linear
stability analysis of the soliton mode. The spectrum and phase shift of the
diffusive modes are determined. In the presence of the soliton the diffusive
modes develop a gap in the spectrum and are phase-shifted in accordance with
Levinson's theorem. The spectrum also exhibits a zero-frequency translation or
Goldstone mode associated with the broken translational symmetry.Comment: 9 pages, Revtex file, 5 figures, to be submitted to Phys. Rev.
The Kardar-Parisi-Zhang equation in the weak noise limit: Pattern formation and upper critical dimension
We extend the previously developed weak noise scheme, applied to the noisy
Burgers equation in 1D, to the Kardar-Parisi-Zhang equation for a growing
interface in arbitrary dimensions. By means of the Cole-Hopf transformation we
show that the growth morphology can be interpreted in terms of dynamically
evolving textures of localized growth modes with superimposed diffusive modes.
In the Cole-Hopf representation the growth modes are static solutions to the
diffusion equation and the nonlinear Schroedinger equation, subsequently
boosted to finite velocity by a Galilei transformation. We discuss the dynamics
of the pattern formation and, briefly, the superimposed linear modes.
Implementing the stochastic interpretation we discuss kinetic transitions and
in particular the properties in the pair mode or dipole sector. We find the
Hurst exponent H=(3-d)/(4-d) for the random walk of growth modes in the dipole
sector. Finally, applying Derrick's theorem based on constrained minimization
we show that the upper critical dimension is d=4 in the sense that growth modes
cease to exist above this dimension.Comment: 27 pages, 19 eps figs, revte
Minimum action method for the Kardar-Parisi-Zhang equation
We apply a numerical minimum action method derived from the Wentzell-Freidlin
theory of large deviations to the Kardar-Parisi-Zhang equation for a growing
interface. In one dimension we find that the switching scenario is determined
by the nucleation and subsequent propagation of facets or steps, corresponding
to moving domain walls or growth modes in the underlying noise driven Burgers
equation. The transition scenario is in accordance with recent analytical
studies of the one dimensional Kardar-Parisi-Zhang equation in the asymptotic
weak noise limit. We also briefly discuss transitions in two dimensions.Comment: 26 pages (revtex) and 18 figures (eps