Time integrators for dispersive equations in the long wave regime

We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical $\varepsilon = 1$ up to long wave limit regime $\varepsilon \ll 1$ on the natural time scale of the PDE $t= \mathcal{O}(\frac{1}{\varepsilon})$. Most notably our new schemes converge with rates at order $\tau \varepsilon$ over long times $t= \frac{1}{\varepsilon}$.