On the propagation of regularity and decay of solutions to the Benjamin equation

In this paper, we investigate some special regularities and decay properties of solutions to the initial value problem(IVP) of the Benjamin equation. The main result shows that: for initial datum $u_0\in H^s(\mathbb{R})$ with $s>3\mathrm{/}4,$ if the restriction of $u_0$ belongs to $H^l((x_0,\mathrm{\infty }))$ for some $l\in {\mathbb{Z}}^{+}$ and $x_0\in \mathbb{R},$ then the restriction of the corresponding solution $u(\cdot ,t)$ belongs to $H^l((\alpha ,\mathrm{\infty }))$ for any $\alpha \in \mathbb{R}$ and any $t\in (0,T)$. Consequently, this type of regularity travels with infinite speed to its left as time evolves.