Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical soluti...

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...

The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow-water waves. Two methods of approach are used. In the first method Boussinesq equations are used to derive evolution equations for spectral-wave components in a slowly varying two-dimensional domain. The second method modifies the K-P equation (Kadomtsev & Petviashvili 1970) to incl...

We construct the traveling wave solutions of the 1 1 -dimensional modified Benjamin-BonaMahony equation, the 2 1 -dimensional typical breaking soliton equation, the 1 1 -dimensional classical Boussinesq equations, and the 2 1 -dimensional Broer-Kaup-Kuperschmidt equations by using an extended G′/G -expansion method, where G satisfies the second-order linear ordinary differential equation. By us...

We prove the existence of a large family of two-dimensional travelling wave patterns for a Boussinesq system which describes three-dimensional water waves. This model equation results from full Euler equations in assuming that the depth of the fluid layer is small with respect to the horizontal wave length, and that the flow is potential, with a free surface without surface tension. Our proof u...

We consider the Stokes-Boussinesq equations in a slanted (that is, not aligned with gravity’s direction) cylinder of any dimension and with an arbitrary Rayleigh number. We prove the existence of a non-planar traveling wave solution, propagating at a constant speed, and satisfying the Dirichlet boundary condition in the velocity and the Neumann condition in the temperature.

In this paper, by using the improved ( ′ G )-expansion method, we have successfully obtained some travelling wave solutions of the variant Boussinesq Equations. These exact solutions include the hyperbolic function solutions, trigonometric function solutions and rational function solutions. Mathematics Subject Classification: 35Q58; 37K50

We give a brief survey of applications of the Gearhart-Prüss spectral mapping theorem for abstract strongly continuous semigroups on Hilbert spaces to the study of stability of solitary waves for a large class of Hamiltonian partial differential equations of mathematical physics including Klein-Gordon, nonlinear Schrödinger, Boussinesq, Benjamin-BonaMahoney (regularized long-wave), Korteweg-deV...

We consider systems of reactive Boussinesq equations in two dimensional strips that are not aligned with gravity’s direction. We prove that for any width of such strips and for arbitrary Rayleigh and Prandtl numbers, the systems admit smooth, non-planar traveling wave solutions with the fluid’s velocity satisfying no-slip boundary conditions.

A composite lattice model is proposed to describe nonlinear waves in a two-layered waveguide with adhesive bonding. We begin by considering waves in an anharmonic chain of oscillating dipoles and show that the corresponding asymptotic long wave model for longitudinal waves coincides with the Boussinesq-type equation, earlier derived for a macroscopic waveguide using the nonlinear elasticity app...