Stochastic stability of master--slave synchronization for dissipative PDEs with Burgers-type nonlinearity

Abstract

We investigate the stochastic stability of master--slave synchronization for a class of nonlinear dissipative PDEs with Burgers-type nonlinearity and polynomial linear operator, including the Burgers, Kuramoto--Sivashinsky, Kawahara, Benney--Lin, and Nikolaevskiy equations. Under periodic boundary conditions, each equation is represented by a finite-dimensional Fourier truncation coupled to a slave driven by observed master data. We establish local exponential stability of the deterministic zero-error synchronization manifold under a simple coupling condition. Introducing observational noise in the coupling transforms the slave into an It\^o diffusion, preventing exact synchronization. We analyze the deviation $\Delta(t)$ between the stochastic error and the exponentially stable deterministic reference error, proving an $\mathcal{O}(\sigma^2)$ finite-time mean-square bound localized near the synchronization manifold, with a tail-probability estimate. Under global one-sided dissipativity, the localization is removed and a time-uniform $\mathcal{O}(\sigma^2)$ bound is obtained. Bounds are derived in Fourier space and transferred to the physical domain via Parseval's identity. Under uniform Sobolev and Galerkin stability assumptions, we also bound the error with respect to the infinite-dimensional master uniformly in truncation order. Results are illustrated by numerical simulations of Kawahara and Nikolaevskiy master--slave systems.