The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation

TL;DR

This paper develops an exact method to analyze large deviations in the KPZ equation at short times by linking it to the integrability of the Zakharov-Shabat system, enabling explicit solutions for various initial conditions.

Contribution

It extends the weak noise theory for the KPZ equation by connecting it to integrable systems, providing a complete solvability framework through inverse scattering techniques.

Findings

Exact solutions for large deviations in KPZ with droplet initial condition.

Connection established between KPZ large deviations and Zakharov-Shabat integrability.

Framework enables future analysis of general initial conditions.

Abstract

We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in one dimension at short time by introducing an approach which combines field theoretical, probabilistic and integrable techniques. We expand the program of the weak noise theory, which maps the large deviations onto a non-linear hydrodynamic problem, and unveil its complete solvability through a connection to the integrability of the Zakharov-Shabat system. Exact solutions, depending on the initial condition of the KPZ equation, are obtained using the inverse scattering method and a Fredholm determinant framework recently developed. These results, explicit in the case of the droplet geometry, open the path to obtain the complete large deviations for general initial conditions.