This paper presents a parametric finite-difference scheme concerning the numerical solution of theone-dimensional Boussinesq-type set of equations, as they were introduced byPeregrine, in the case of wavesrelatively long with small amplitudes in water of constant depth. The method which is used can be considered asa generalization of the Crank-Nickolson method and it has been ap...

In this paper, an improved scheme is presented to investigate the approximate solutions of (2+1)-dimensional Boussinesq equation with initial conditions based on the decomposition method. The improve scheme makes all components ui of u = ∑∞ i=0 ui be of the same degree in t. We choose the (2+1)-dimensional Boussinesq equation with initial conditions of doubly-periodic functions to illustrate th...

In this paper, we study the possible second order Lax operators for all the possible (1+1)-dimensional models with Schwarz variants and some special types of high dimensional models. It is shown that for every (1+1)-dimensional model and some special types of high dimensional models which possess Schwarz variants may have a second order Lax pair. The explicit Lax pairs for (1+1)-dimensional Kor...

The pentagram map is a projectively natural transformation defined on (twisted) polygons. A twisted polygon is a map from Z into RP that is periodic modulo a projective transformation called the monodromy. We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable. For certain families of twisted polyg...

The Boussinesq system arises in Fluid Mechanics when motion is governed by density gradients caused by temperature or concentration di erences. In the former case, and when thermodynamical coe cients are regarded as temperature dependent, the system consists of the Navier-Stokes equations and the non linear heat equation coupled through the viscosity, bouyancy and convective terms. In this pape...

We develop a robust Christov–Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in Lð 1;1Þ is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which...

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the const...

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...