We present a new algorithm for the rigorous integration of the variational equation (i.e. producing $\mathcal C^1$ estimates) for a class of dissipative PDEs on the torus. As an application for some parameter value for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions we prove the existence of infinite number of homo- and heteroclinic orbits to two periodic orbits. The proof is computer assisted.

翻译：本文提出了一种用于严格积分环面上某类耗散偏微分方程变分方程（即生成$\mathcal C^1$估计）的新算法。作为应用，针对具有奇周期边界条件的线性Kuramoto-Sivashinsky偏微分方程的某个参数值，我们证明了存在无穷多条连接两个周期轨道的同宿与异宿轨道。该证明为计算机辅助证明。