Krajenbrink Alexandre, Le Doussal Pierre
SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy and Quantinuum and Cambridge Quantum Computing, Cambridge, United Kingdom.
Laboratoire de Physique de l'École Normale Supérieure, CNRS, ENS and PSL University, Sorbonne Université, Université de Paris, 75005 Paris, France.
Phys Rev E. 2022 May;105(5-1):054142. doi: 10.1103/PhysRevE.105.054142.
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 nonlinear hydrodynamic equations of the WNT are solved analytically through a connection 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 nonvanishing 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.
我们给出了一维 Kardar-Parisi-Zhang 方程在短时间内针对平坦初始条件（IC）的弱噪声理论（WNT）的解。通过利用其经典可积性与 Zakharov-Shabat（ZS）系统建立联系，解析求解了 WNT 的非线性流体动力学方程。这种方法基于最近开发的 Fredholm 行列式框架，该框架先前已应用于液滴初始条件。平坦初始条件为 ZS 系统提供了一个非零边界条件的情形，并产生了更丰富的孤子结构，包括兰伯特函数多个分支的出现。作为副产品，我们得到了经历动力学相变的布朗初始条件下 WNT 的显式解。我们通过表明相关的空间对称性自发破缺源于两个具有不同速度的孤子之间的相互作用来阐明其机制。
