We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirectional solutions of the improved Boussinesq equation tend to associated solutions of two uncoupled Camassa-Holm equations. We give a precise estimate for approximation errors in terms of two small positive parameters measuring the effects of nonlinearity and disp...

We consider the (KdV)/(KP-I) asymptotic regime for the Nonlinear Schrödinger Equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg-de Vries equation (in dimension 1) and to the Kadomtsev-Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/...

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin-Bona-Mahony type equations posed on a bounded interval. We study the stabilization of the resulting system when a localized damp...

where p = p(x, t) is the scalar pressure, u = u(x, t) is the velocity vector field, and θ = θ (x, t) is a scalar quantity such as the concentration of a chemical substance or the temperature variation in a gravity field, in which case θe represents the buoyancy force. The nonnegative parameters μ and ν denote the molecular diffusion and the viscosity, respectively; u · ∇u := ua∂au, ∂au := ∂u ∂...

In this article, we establish new traveling wave solutions for the loaded Boussinesq equation and modified by functional variable method. The performance of method is reliable effective gives exact solitary periodic its modifications. obtained via are expressed hyperbolic functions trigonometric functions. graphical representations some demonstrated to better understand their physical features,...

Spiral-defect chaos (SDC) in Rayleigh-Bénard convection is a well-established spatio-temporal complex pattern, which competes with stationary rolls near the onset of convection. The characteristic properties of SDC are accurately described on the basis of the standard three-dimensional Boussinesq equations. As a much simpler and attractive two-dimensional model for SDC generalized Swift-Hohenbe...

In this paper a new set of Boussinesq wave equations with improved linear dispersion properties is proposed for extending its application to deeper water without having its mathematical form changed. The improvements are due to the combination of a generalized set of Boussinesq wave equations expressed in terms of any velocity, with additional dispersive terms obtained by invoking the linear sh...

In this paper, we focus on the implementation and verification of an extended Boussinesq model for surf zone hydrodynamics in two horizontal dimensions. The time-domain numerical model is based on the fully nonlinear Boussinesq equations. As described in Part I of this two-part paper, the energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term into the momentum ...

New nonlinear evolution equations are derived that generalize those presented in a Letter by Matsuno [Phys. Rev. Lett. 69, 609 (1992)]] and a terrain-following Boussinesq system recently deduced by Nachbin [SIAM J Appl. Math. 63, 905 (2003)]]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. A F...

Parts I and II of this paper describe the extension and testing of two sets of Boussinesq-type equations to include surf zone phenomena. Part I is restricted to 1D tests of shoaling, breaking, and runup, while Part II deals with two horizontal dimensions. The model uses two main extensions to the Boussinesq equations: a momentum-conserving eddy viscosity technique to model breaking, and a ‘‘slo...