_{1}

^{*}

The main aim of this work is to introduce the analytical approximate solutions of the water wave problem for a fluid layer of finite depth in the presence of gravity. To achieve this aim, we begun with the derivation of the Korteweg-de Vries equations for solitons by using the method of multiple scale expansion. The proposed problem describes the behavior of the system for free surface between air and water in a nonlinear approach. To solve this problem, we use the well-known analytical method, namely, variational iteration method (VIM). The proposed method is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. The proposed method provides a sequence of functions which may converge to the exact solution of the proposed problem. Finally, we observe that the elevation of the water waves is in form of traveling solitary waves.

We are concerned with a two-dimensional, irrotational flow of an incompressible ideal fluid with a free surface under the gravitational field. The domain occupied by the fluid is bounded from below by a solid bottom. The upper surface is a free boundary, so we take the influence of the gravitational field into account on the free surface. Our main interest is motion of the free surface, which is called a gravity wave.

The Korteweg-de Vries equation (KdV) was originally derived by Korteweg and de Vries from the model surface waves in a canal. The key to a soliton’s behavior is a robust balance between the effects of dispersion and nonlinearity. When one grafts these two effects onto the wave equation for shallow water waves, then at leading order in the strengths of the dispersion and nonlinearity one gets the KdV for solitons [

The Korteweg-de Vries equation was obtained by Benjamin [

where X stands for a space coordinate, T denotes time, and Z represents the surface elevation of a liquid in a shallow duct. This equation can be derived perturbatively from the Euler equation for the motion of an incompressible and inviscid fluid [

which describes a hump of invariable shape moving to the right with velocity c for all values of

Concerning the Korteweg-de Vries approximation for water waves, we refer to Craig [

Many different methods have recently introduced to solve nonlinear problems such as, VIM [

The structure of the paper is organized as follows. In Section 2, we collect the basic equations and boundary conditions. In Section 3, the analysis of the VIM is introduced. In Section 4, we implement VIM to investigate the two-soliton solutions for Korteweg-de Vries equations. Finally, Section 5 contains some conclusions.

We consider the unsteady two-dimensional flow of inviscid, incompressible fluid in a constant gravitational field. The space coordinates are (x, y) and the gravitational acceleration g is in the negative y direction. Let h be the undisturbed depth of the fluid. The bottom of the fluid is assumed to have no topography at

This problem describes the interface dynamics, between air and water waves, under the gravity g (see

The equation for the surface of water is

The divergence-free condition on the velocity field implies that the velocity potential

On a solid fixed boundary, the normal velocity of the fluid must vanish:

which dictates that there is no flow perpendicular to the bottom.

The boundary conditions at the free surface

Now we try to find the approximate solutions by using VIM of following Korteweg-de Vries equations (which are derived in details in [

To illustrate the analysis of VIM [

with specified initial conditions, where L and R are linear bounded operators, and

It is obvious that the successive approximations

In what follows, we will apply VIM to Korteweg-de Vries Equations (8)-(9) to illustrate the strength of the method and to obtain the approximate solutions for this nonlinear problem.

Now, to illustrate how to find the value of the Lagrange multiplier

Making the above correction functional stationary, and noticing that

where

The first equation in (13) is called Lagrange-Euler equation and the second equation in (13) is called natural boundary condition, the Lagrange multiplier, therefore, can be readily identified,

Now, the following variational iteration formula can be obtained:

We start with an initial approximation, and by using the above iteration formula (14), we can obtain directly the other components of the solution.

In this section, we will implement VIM to Korteweg-de Vries Equations (8)-(9), with the initial conditions:

To solve Equations (8)-(9) by means of VIM, we construct the correction functionals which read:

Making the above correction functional stationary, and noticing that

where

The Lagrange multipliers

Now, the following variational iteration formula can be obtained:

We start with an initial approximations

Returning to dimensional variables, we get:

In the same manner, we can obtain other components of the solution. In order to verify numerically whether the proposed methodology lead to higher accuracy, we can evaluate the numerical solutions using

From the above solution process, we can see clearly that the approximate solutions converge to its exact solution relatively slowly due to the approximate identification of the multiplier. It should be specially pointed out that the more accurate the identification of the multiplier, faster the approximations converge to its exact solutions.

After returning to dimensional variables and substitution from (27) into (3) we get the elevation of the water surface, the horizontal velocity, the vertical velocity and the phase diagrams of the velocity, which describes the physical situation of the system, where:

The water wave gradually splits into two solitary waves with increasing

In this study, we present model equations for surface water waves by using a new method of multiple scale technique. Multiple scale technique is used to estimate the Korteweg-de Vries equations for the nonlinear theory, describing the behaviour of the perturbed system. We observed that the method of multiple scale was one

of the modern methods which we used to obtain the Korteweg-de Vries equations because it was relatively short in mathematical calculation, more effective and more enlightening. While the Hamiltonian expansions and the Dirichlet-Neumann operator expansions are complex in mathematical calculation and relatively long method

as compared with the method of multiple scale.

The diagrams are drawn to illustrate the elevation of the water waves that show a solitary character. We observed that the elevation of the water waves is in form of traveling solitary waves, which increases in amplitude as the wave number increases.

Finally, the horizontal and vertical velocities of the velocity components have nonlinear characters, which describes the physical situation of the system for free surface between air and water.

The presented examples show that the results of the proposed method VIM are in excellent agreement with the exact solution. An interesting point about VIM is that only few iterations or, even in some special cases, one iteration, lead to exact solutions or solutions with high accuracy. The main merits of VIM are:

1) VIM can overcome the difficulties arising in calculation of Adomian’s polynomials in Adomain decomposition method.

2) VIM does not require small parameters which are needed in perturbation method.

3) No linearization is needed; the method is very promising for solving wide application in nonlinear differential equations.

In our work, we used the Mathematica Package.

The author thanks the Editor and the referee for their comments.

Rabab Fadhel Al-Bar, (2016) Numerical Simulation for Nonlinear Water Waves Propagating along the Free Surface. Journal of Applied Mathematics and Physics,04,930-938. doi: 10.4236/jamp.2016.45102