Inverse Scattering of the Zakharov-Shabat System Solves the Weak Noise Theory of the Kardar-Parisi-Zhang Equation

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 nonline...

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Bibliographic Details
Published in:Physical review letters 2021-08, Vol.127 (6), p.064101-064101, Article 064101
Main Authors: Krajenbrink, Alexandre, Le Doussal, Pierre
Format: Article
Language:English
Subjects:
Data Analysis, Statistics and Probability
Deviation
Exact solutions
Initial conditions
Inverse scattering
Physics
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Summary: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 nonlinear 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.
ISSN:0031-9007
1079-7114
DOI:10.1103/PhysRevLett.127.064101