Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions

TL;DR

This paper analytically solves the weak noise theory for the 1D KPZ equation at short times with flat and Brownian initial conditions, revealing complex solitonic structures and a dynamical phase transition.

Contribution

It introduces an analytical solution to the WNT for KPZ with flat and Brownian initial conditions using integrability and Fredholm determinants, uncovering new solitonic phenomena.

Findings

Explicit solution for flat initial condition with rich solitonic structure.

Identification of a dynamical phase transition for Brownian initial condition.

Spontaneous symmetry breaking explained by soliton interactions.

Abstract

We present the solution of the weak noise theory (WNT) for the Kardar-Parisi-Zhang equation in one dimension at short time for flat initial condition (IC). The non-linear hydrodynamic equations of the WNT are solved analytically through a connexion to the Zakharov-Shabat (ZS) system using its classical integrability. This approach is based on a recently developed Fredholm determinant framework previously applied to the droplet IC. The flat IC provides the case for a non-vanishing boundary condition of the ZS system and yields a richer solitonic structure comprising the appearance of multiple branches of the Lambert function. As a byproduct, we obtain the explicit solution of the WNT for the Brownian IC, which undergoes a dynamical phase transition. We elucidate its mechanism by showing that the related spontaneous breaking of the spatial symmetry arises from the interplay between two solitons with different rapidities.