Solitary wave solutions, periodic and superposition solutions to the system of first-order (2+1)-dimensional Boussinesq's equations derived from the Euler equations for an ideal fluid model

TL;DR

This paper investigates (2+1)-dimensional nonlinear wave equations from ideal fluid models, deriving solutions like solitary, periodic, and superposition waves from Boussinesq equations.

Contribution

It introduces new (2+1)-dimensional wave solutions to Boussinesq equations derived from Euler equations for ideal fluids.

Findings

Existence of families of traveling wave solutions including solitary and periodic waves.

Solutions include cnoidal and superposition types.

Surface wave forms can be obtained from solutions of an auxiliary function.

Abstract

This article concludes the study of (2+1)-dimensional nonlinear wave equations that can be derived in a model of an ideal fluid with irrotational motion. In the considered case of identical scaling of the $x,y$ variables, obtaining a (2+1)-dimensional wave equation analogous to the KdV equation is impossible. Instead, from a system of two first-order Boussinesq equations, a non-linear wave equation for the auxiliary function $f(x,y,z)$ defining the velocity potential can be obtained, and only from its solutions can the surface wave form $\eta(x,y,t)$ be obtained. We demonstrate the existence of families of (2+1)-dimensional traveling wave solutions, including solitary and periodic solutions, of both cnoidal and superposition types.