Soliton approach to the noisy Burgers equation: Steepest descent method

The noisy Burgers equation in one spatial dimension is analyzed by means of the Martin-Siggia-Rose technique in functional form. In a canonical formulation the morphology and scaling behavior are accessed by means of a principle of least action in the asymptotic nonperturbative weak noise limit. The ensuing coupled saddle point field equations for the local slope and noise fields, replacing the noisy Burgers equation, are solved yielding nonlinear localized soliton solutions and extended linear diffusive mode solutions, describing the morphology of a growing interface. The canonical formalism and the principle of least action also associate momentum, energy, and action with a soliton-diffusive mode configuration and thus provide a selection criterion for the noise-induced fluctuations. In a ``quantum mechanical'' representation of the path integral the noise fluctuations, corresponding to different paths in the path integral, are interpreted as ``quantum fluctuations'' and the growth morphology represented by a Landau-type quasiparticle gas of ``quantum solitons'' with gapless dispersion E∝P3/2 and ``quantum diffusive modes'' with a gap in the spectrum. Finally, the scaling properties are discussed from a heuristic point of view in terms of a ``quantum spectral representation'' for the slope correlations. The dynamic exponent $z=3/2$ is given by the gapless soliton dispersion law, whereas the roughness exponent ζ=1/2 follows from a regularity property of the form factor in the spectral representation. A heuristic expression for the scaling function is given by a spectral representation and has a form similar to the probability distribution for L\'evy flights with index $z.$