In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different ...

In this report, generalized wave breaking equations are developed using three dimensional fully nonlinear extended Boussinesq equations to encompass rotational dynamics in wave breaking zone. The derivation for vorticity distributions are developed from Reynold based stress equations.

Zufiria’s higher-order Boussinesq type equations are studied by transforming them into solvable ordinary differential equations. Various families of their travelling wave solutions are generated, which include periodic wave, solitary wave, periodic-like wave, solitonlike wave, Jacobi elliptic function periodic wave, combined non-degenerative Jacobi elliptic function-like wave, Weierstrass ellip...

An improved curvilinear grid model based on fully nonlinear Boussinesq equations is used to simulate wave propagation in Ponce de Leon Inlet, Fla. We employ the nearshore bathymetry of Ponce de Leon Inlet and generate a stretched curvilinear grid that can resolve shortwaves in the nearshore region and fit the complex geometry. Simulations of 18 cases with monochromatic input waves and Texel-Mar...

In inviscid fluid flows instability arises generically due to a resonance between two wave modes. Here, it is shown that the structure of the weakly nonlinear régime depends crucially on whether the modal structure coincides, or remains distinct, at the resonance point where the wave phase speeds coincide. Then in the weakly nonlinear, long-wave limit the generic model consists either of a Bous...

using the mean-value theorem for integrals we tried to solved the nonlinear integral equations of hammerstein type . the mean approach is to obtain an initial guess with unknown coefficients for unknown function y(x). the procedure of this method is so fast and don't need high cpu and complicated programming. the advantages of this method are that we can applied for those integral equation...

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In par...

We apply the concepts of nonlinear guided-wave optics to a Bose-Einstein condensate ~BEC! trapped in an external potential. As an example, we consider a parabolic double-well potential and derive coupled-mode equations for the complex amplitudes of the BEC macroscopic collective modes. Our equations describe different regimes of the condensate dynamics, including the nonlinear Josephson effect ...

The long wave–short wave resonance model arises physically when the phase velocity of a long wave matches the group velocity of a short wave. It is a system of nonlinear evolution equations solvable by the Hirota bilinear method and also possesses a Lax pair formulation. ‘‘Rogue wave’’ modes, algebraically localized entities in both space and time, are constructed from the breathers by a singul...

We describe a pseudo-spectral numerical method to solve the systems of one-dimensional evolution equations for free surface waves in a homogeneous layer of an ideal fluid. We use the method to solve a system of one-dimensional integro-differential equations, first proposed by Ovsjannikov and later derived by Dyachenko, Zakharov, and Kuznetsov, to simulate the exact evolution of nonlinear free s...