3 research outputs found
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.
Canonical phase space approach to the noisy Burgers equation: Probability distributions
We present a canonical phase space approach to stochastic systems described
by Langevin equations driven by white noise. Mapping the associated
Fokker-Planck equation to a Hamilton-Jacobi equation in the nonperturbative
weak noise limit we invoke a {\em principle of least action} for the
determination of the probability distributions. We apply the scheme to the
noisy Burgers and KPZ equations and discuss the time-dependent and stationary
probability distributions. In one dimension we derive the long-time skew
distribution approaching the symmetric stationary Gaussian distribution. In the
short-time region we discuss heuristically the nonlinear soliton contributions
and derive an expression for the distribution in accordance with the directed
polymer-replica and asymmetric exclusion model results. We also comment on the
distribution in higher dimensions.Comment: 18 pages Revtex file, including 8 eps-figures, submitted to Phys.
Rev.