Close to the origin, the nonlinear Klein--Gordon equations on the circle are nearly integrable Hamiltonian systems which have infinitely many almost conserved quantities called harmonic actions or super-actions. We prove that, at low regularity and with a CFL number of size 1, this property is preserved if we discretize the nonlinear Klein--Gordon equations with the symplectic mollified impulse methods. This extends previous results of D. Cohen, E. Hairer and C. Lubich to non-smooth solutions.

翻译：在原点附近，圆周上的非线性 Klein--Gordon 方程是近似可积的哈密顿系统，其具有无穷多个几乎守恒量，称为谐波作用量或超作用量。本文证明，在低正则性条件下且 CFL 数约为 1 时，若采用辛格式的平滑冲量方法对非线性 Klein--Gordon 方程进行离散化，该性质依然得以保持。这一结果将 D. Cohen、E. Hairer 和 C. Lubich 先前的研究推广至非光滑解的情形。