On the Benjamin and related equations

TL;DR

This paper investigates classical one-dimensional internal wave models with surface tension, exploring theoretical and numerical aspects such as long-term dynamics, solitons, and solution behaviors, highlighting differences caused by capillary effects.

Contribution

It surveys existing results, introduces new findings, and discusses open questions on the impact of surface tension on wave equations, emphasizing qualitative solution differences.

Findings

Analysis of long-time dynamics of solutions

Identification of soliton and multisoliton behaviors

Differences in solution behavior due to capillary effects

Abstract

We consider in this paper various theoretical and numerical issues on classical one dimensional models of internal waves with surface tension.They concern the Cauchy problem, including the long time dynamic, localized solitons or multisolitons, the soliton resolution property. We survey known results, present a few new ones together with open questions and conjectures motivated by numerical simulations. A major issue is to emphasize the differences of the qualitative behavior of solutions with those of the same equations without the capillary term.