Using the algebraic method of Gardner’s deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup–Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry proper...

Exact solutions of the dispersive water wave and modified Boussinesq equations are expressed in terms of special polynomials associated with rational solutions of the fourth Painlevé equation, which arises as generalized scaling reductions of these equations. Generalized solutions that involve an infinite sequence of arbitrary constants are also derived which are analogues of generalized ration...

We study the problem of gravity surface waves for an ideal fluid model in (2+1)-dimensional case. apply a systematic procedure to derive Boussinesq equations given relation between orders four expansion parameters, amplitude parameter $\alpha$, long-wavelength $\beta$, transverse wavelength $\gamma$, and bottom variation $\delta$. derived only possible extensions Korteweg-de Vries equation, fif...

Abstract A new three dimensional nonlinear dynamic theoretical model is derived from fluid mechanics system. In this paper, From the quasi-geostrophic barotropic potential vorticity equation, we obtain a dissipative Boussinesq equation by reduced perturbation method, i.e. u tt + e 1 xx 2 ( ) 3 txy 4 xxxx 5 xxyy = 0 . It emphasized that different existing equations, which describe Rossby waves i...

The paper considers a system of equations that models lateral flow Boussinesq–Scriven fluid on passively evolving surface embedded in [Formula: see text]. For the resulting Navier–Stokes type system, posed smooth closed time-dependent surface, we introduce weak formulation terms functional spaces space-time manifold defined by evolution. is shown to be well-posed for any finite final time and w...

In this paper we study finite dimensional approximations to Boussinesq type equations. Our methods are based on infinite dimensional center manifold theory. The main advantage of our approach is that we can handle both well-posed and ill-posed versions of the Boussinesq equation. We show that for suitable initial conditions, our approximations describe the dynamics accurately for long enough ti...

A family of Boussinesq systems has recently been proposed by Bona, Chen, and Saut in [J.L. Bona, M. Chen, J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory, J. Nonlinear Sci. 12 (4) (2002) 283–318] to describe the two-way propagation of small-amplitude gravity waves on the surface of water in a canal....

For flows with small density variations, it is convenient to approximate them as incompressible whilst retaining the leading order effects due to the density variations, thus avoiding issues associated with acoustic waves. The classical approach is the Boussinesq approximation which was originally motivated by the desire to account for gravitational buoyancy effects. There are many situations o...

Intermediate-depth, Boussinesq-type modelling is used to generalize previously known results for surface water waves propagating over arbitrarily shaped topographies. The improved reduced wave model is obtained after studying how small changes in the linear dispersion relation (over a flat bottom) can become dramatically important in the presence of a highly fluctuating topography. Numerical va...

This paper describes a σ -coordinate scalar transport model coupled with a Boussinesq-type hydrodynamic model. The Boussinesq model has the ability to calculate both three-dimensional velocity distributions and the water surface motion. To capture ‘dispersion’ processes in open channel flow, horizontal vorticity effects induced by a bottom shear stress are included in the Boussinesq model. Thus...