A Boussinesq-type wave model is developed to numerically investigate the breaking waves and wave-induced currents. All the nonlinear terms are retained in the governing equations to keep fully nonlinearity characteristics and it hence more suitable to describe breaking waves with strong nonlinearity in the nearshore region. The Boussinesq equations are firstly extended to incorporate wave break...

This paper describes the numerical model BOUSS-WMH (BOUSSinesq Wave Model for Harbours), a finite element model for nonlinear wave propagation near shore and into harbors. It is based upon an extended version of the Boussinesq equations to which terms were added to generate regular or irregular waves inside the numerical domain, absorb outgoing waves, partially reflect waves at physical boundar...

Measurements of the Nusselt number N as a function of the Rayleigh number R using ethane gas near its critical point show that N is enhanced by non-Oberbeck-Boussinesq (NOB) effects when the top temperature approaches the coexistence curve. The empirical criterion that a small ∆ρ/ρ (ρ is the density) implies Boussinesq behavior is found not to be valid. An examination of data by Chavanne et al....

Boussinesq systems of nonlinear partial differential equations are fundamental equations in geophysical fluid dynamics. In this paper, we use asymmetric ideas and moving frames to solve the two-dimensional Boussinesq equations with partial viscosity terms studied by Chae (Adv. Math. 203 (2006), 497-513) and the three-dimensional stratified rotating Boussinesq equations studied by Hsia, Ma and W...

A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z. The models that result from this framework are Hamiltonian, and have a Kelvin– Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler– Boussinesq equations with a constant t...

We construct new integrable coupled systems of N = 1 supersymmetric equations and present integrable fermionic extensions of the Burgers and Boussinesq equations. Existence of infinitely many higher symmetries is demonstrated by the presence of recursion operators. Various algebraic methods are applied to the analysis of symmetries, conservation laws, recursion operators, and Hamiltonian struct...

The modulational stability of both the Korteweg-de Vries (KdV) and the Boussinesq wavetralns is investigated using Whltham’s variational method. It is shown that both KdV and Boussinesq wavetrains are modulationally stable. This result seems to confirm why it is possible to transform the KdV equation into a nonlinear Schr’dinger equation with a repulsive potential. A brief discussion of Whltham...

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...

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...