Mathematical modeling and numerical analysis for the higher order Boussinesq system

TL;DR

This paper develops and analyzes higher-order Boussinesq equations for water waves, proving well-posedness, comparing solutions with actual water wave models, and validating the model through explicit solutions and numerical simulations.

Contribution

It introduces a higher-order Boussinesq model for water waves, establishes its well-posedness, and provides numerical validation and comparison with real water wave solutions.

Findings

The model is well-posed in the Hadamard sense.

Solutions of the model closely approximate water wave solutions.

Numerical validation confirms the model's accuracy.

Abstract

This study deals with higher-ordered asymptotic equations for the water-waves problem. We considered the higher-order/extended Boussinesq equations over a flat bottom topography in the well-known long wave regime. Providing an existence and uniqueness of solution on a relevant time scale of order $1/\sqrt{\eps}$ and showing that the solution's behavior is close to the solution of the water waves equations with a better precision corresponding to initial data, the asymptotic model is well-posed in the sense of Hadamard. Then we compared several water waves solitary solutions with respect to the numerical solution of our model. At last, we solve explicitly this model and validate the results numerically.