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

Understand

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.