We investigate the interaction of a small-amplitude internal gravity wave packet in a rotating fluid with a baroclinic shear flow. Ray equations and three-dimensional direct numerical simulations of the Boussinesq equations are solved for this purpose. The shear flow has a simple structure and depends separately on the vertical and horizontal coordinates. We focus on the situation where the int...

The effects of randomly irregular bathymetry on the propagation of interfacial gravity waves are studied. Modelling sea water by a two-layer fluid of different densities, weakly nonlinear waves much longer than the sea depth but comparable to the horizontal scale of bathymetry are treated by Boussinesq approximation and multiplescale analysis. For transient wave pulses, the governing equation f...

We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in Hm with m ≥ 3. It is known that when both the velocity and the density equations have finite positive viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the veloci...

New results concerning the orbital stability of periodic traveling wave solutions for 'abcd' Boussinesq model will be shown in this manuscript. For existence solutions, we use basic tools ordinary differential equations to show that corresponding depends on Jacobi elliptic function cnoidal type. The spectral analysis associated linearized operator is determined by using some Floquet theory. the...

In this paper, we investigate the new traveling wave solutions for sixth-order Boussinesq equation using tanh-coth method. Such a method is type of expansion that represents partial differential equations as polynomials $\tanh$ and $\coth$ functions. We discover several with different parameters, which are fundamental importance various applications.

This contribution concerns a specific simulation method for coastal wave engineering applications. As is common to reduce computational costs the flow is assumed to be irrotational so that a Boussinesq-type of model in horizontal variables only can be used. Here we advocate the use of such a model that respects the Hamiltonian structure of the wave equations. To avoid approximations of the disp...

and Applied Analysis 3 Two papers in our special issue are concerned about the study of application to the symbolic computation method and special function. In the paper titled “New nonlinear systems admitting Virasoro-type symmetry algebra and groupinvariant solutions” by L. Wang et al., the authors extend the application of Virasoro-type symmetry prolongationmethod to coupled systems with two...

The Serre–Green–Naghdi (SGN) equations, also known as the fully-nonlinear Boussinesq wave equations, accurately describe the behaviour of dispersive shoaling water waves. This article presents and validates a novel combination of methods for the numerical approximation of solutions to the SGN equations. The approach preserves the robustness of the original finite-volume Saint-Venant solver, in ...

It is well known that the Boussinesq equation is the bidirectional equivalent of the celebrated Korteweg-de Vries equation. Here we consider Boussinesq-type versions of two classical unidirectional integrable equations. A procedure is presented for deriving multisoliton solutions of one of these equations – a bidirectional Kaup–Kupershmidt equation. These solitons have the unusual property that...

Abstract Many of the widely used models for description nonlinear surface gravity waves, in deep or shallow water, such as High Order Spectral Method (HOSM) and Boussinesq-type equations, rely on elimination vertical coordinate from basic three-dimensional Euler equations. From a numerical point view are often computationally efficient, which is one main reasons that many frequently studies wav...