Aspects of the noisy burgers equation

Abstract

Abstract. 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 MartinSiggia-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 E ∝ p 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.

Langevin equations for continuous time Lévy flights

“…From a physical point of view it is clear that a nontrivial waiting time distribution has no effect since the quenched force field acting at position r simply 'waits' till the walker arrives. However, in the case of a time dependent random force field the waiting time distribution will interfere with the temporal force correlations and we have to treat the coupled Langevin equations in order to eliminate the intermediate path variable s. This interesting case will be considered in a forthcoming publication [22].…”

Morphology and Scaling in the Noisy Burgers Equation: Soliton Approach to the Strong Coupling Fixed Point

“…It is well-known that the noiseless Burgers equation for ∆ = 0 [1,4,6], ∂u/∂t = ν∇ 2 u + λu∇u, supports paritybreaking right hand solitons connected by ramp solutions, corresponding to cusps connected by convex parabolic segments in the height field. Superposed on the solitons are linear diffusive modes with a gap in the spectrum.…”

“…The linear diffusive mode spectrum in the presence of the solitons is analyzed by a linear stability analysis of the field equations (13). Like in the noiseless case [6] the associated eigenvalue problem is exactly soluble [12]. In addition to zero-eigenvalue bound states corresponding to the soliton translation modes lifting the broken translational symmetry, the spectrum also has a band of phase shifted diffusive modes with a gap in the dispersion law…”