We establish improved uniform error bounds on a second-order Strang time-splitting method which is equivalent to an exponential wave integrator for the long-time dynamics of nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength characterized by $\varepsilon^2$ $0 < \varepsilon \leq 1$ dimensionless parameter. Actually, when \ll 1$, NKGE $O(\varepsilon^2)$ nonlinea...

We study absorbing phase transitions in the one-dimensional branching annihilating random walk with long-range repulsion. The repulsion is implemented as hopping bias such a way that particle more likely to hop away from its closest particle. strength due interaction has form $\varepsilon x^{-\sigma}$, where $x$ distance particle, $0\le \sigma \le 1$, and sign of $\varepsilon$ determines whethe...