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

TL;DR

This paper studies how regularity and decay properties of solutions to the Benjamin equation evolve over time, showing that certain regularities propagate infinitely fast to the left, depending on initial conditions.

Contribution

It establishes new results on the propagation of regularity and decay for solutions to the Benjamin equation with specific initial data.

Findings

Regularity propagates infinitely fast to the left over time.

Solutions with initial regularity in a half-line maintain and extend this regularity.

Decay properties are linked to initial data in Sobolev spaces.

Abstract

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 $s>3/4,$ if the restriction of $u_{0}$ belongs to $H^{l}((x_{0}, \infty))$ for some $l\in \mathbb{Z}^{+}$ and $x_{0}\in \mathbb{R},$ then the restriction of the corresponding solution $u(\cdot, t)$ belongs to $H^{l}((\alpha, \infty))$ 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.