Symmetries of the stochastic Burgers equation

TL;DR

This paper identifies all Lie symmetries of the stochastic Burgers equation, revealing invariances and their breaking by Gaussian forcing, and connects symmetry analysis to an exactly solvable quantum problem.

Contribution

It uncovers the full symmetry structure of the stochastic Burgers equation and links symmetry breaking to the formation of a nontrivial vacuum state.

Findings

Identified all Lie symmetries including generalized Galilean and time reparametrizations.

Showed Gaussian forcing breaks these symmetries, leading to instanton solutions.

Mapped the symmetry analysis to an exactly solvable quantum mechanical problem.

Abstract

All Lie symmetries of the Burgers equation driven by an external random force are found. Besides the generalized Galilean transformations, this equation is also invariant under the time reparametrizations. It is shown that the Gaussian distribution of a pumping force is not invariant under the symmetries and breaks them down leading to the nontrivial vacuum (instanton). Integration over the volume of the symmetry groups provides the description of fluctuations around the instanton and leads to an exactly solvable quantum mechanical problem.