Long time existence for a strongly dispersive Boussinesq system

TL;DR

This paper proves long-time existence of solutions for a highly dispersive one-dimensional Boussinesq system using advanced normal form techniques, contributing to the understanding of dispersive PDEs.

Contribution

It introduces two novel long-time existence results for a specific strongly dispersive Boussinesq system, employing modified normal form transformations.

Findings

Established long-time existence for solutions of the system.

Developed modified normal form methods for dispersive PDEs.

Enhanced understanding of dispersive Boussinesq systems.

Abstract

This paper is concerned with the one-dimensional version of a specific member of the (abcd) family of Boussinesq systems having the higher possible dispersion. We will establish two different long time existence results for the solutions of the Cauchy problem. The proofs involve normal form transformations suitably modified away from the zero set of the phases.