Boussinesq equation

TL;DR

This paper discusses the Boussinesq equation, a fundamental model for long wave motion in two dimensions, highlighting its well-posedness depending on parameters and emphasizing the importance of external interactions in physical waveguide studies.

Contribution

It provides an overview of the Boussinesq equation's properties, including well-posedness conditions and the significance of external energy exchange in nonlinear wave analysis.

Findings

Well-posedness depends on parameter c (-1 is well-posed, 1 is not)

External environment interactions influence wave deformation in viscous media

Energy exchange impacts wave dispersion in waveguides

Abstract

Boussinesq equation belongs to Korteweg-de Vries kind of equations (Han & Yarkony, 2011). Equation describes the motion of long waves in two dimensions under the gravitation (Han & Yarkony, 2011). Here, we differentiate u = u(x, t) to the needed order. With c=-1 we have well-posed Boussinesq equation, while with c=1 not well-posed classical Boussinesq equation (Gradinaru, Hagedorn, & Joye, 2010). In physical research of nonlinear waves spread in waveguides, it is necessary to take into account the interaction of waveguides with external environment (Berezhkoyskii, 2011). Therefore, the possibility of energy exchange through the outer surface of waveguides, is also should be taken into account (Berezhkoyskii, 2011). When the energy exchange between a rod and external environment occurs, it is important to estimate the deformation dispersion of wave in viscous environment (Han & Yarkony, 2011).