In this paper, we consider a class of evolution equations driven by finite-dimensional $$\gamma $$ -Hölder rough paths, where \in (1/3,1/2]$$ . We prove the global-in-time solutions (REEs) in sutiable space, also obtain that generate random dynamical systems. Meanwhile, derive existence local unstable manifolds for such properly discretized Lyapunov–Perron method.

<p style='text-indent:20px;'>We outline the construction of special functions in terms Fredholm determinants to solve boundary value problems string spectral problem. Our motivation is that problem related equations Lax pairs at least three nonlinear evolution from mathematical physics.</p>