Discrete equations and auto-traveling kinks of the $\phi^6$ model

We study the $\phi^{6}$ model and derive two broad classes of lattice discretizations that admit static, translationally invariant kinks; that is, stationary kink profiles that can be centered at an arbitrary position relative to the lattice. These discretizations are constructed using a one-dimensional map, $\phi_{n+1}=F(\phi_{n})$, which provides a direct and systematic algorithm for generating such models. Numerical computations for two representative cases show that the discrete kinks do not possess internal modes, consistent with the continuum theory, although an additional high-frequency mode may appear above the phonon band. We also show that generic discretizations of the $\phi^{6}$ model do not support static kink solutions. Instead, the resulting dynamics produce auto-traveling and self-accelerating kinks that propagate at the maximal group velocity while continuously emitting radiation.