Solitary Wave Experiment

Solitons
“True laws of Nature cannot be linear. - Albert Einstein”
Introduction:
One of the most important topics at the beginning of the nineteenth century was the work of John Russell on the water wave mechanics. On his “Report on waves”, Russel was able to discover the solitary wave phenomenon [1], which happened near Edinburgh, Scotland. “ I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary

Later on, his argument was accepted by universities, which resulted in finding equations and solutions of nonlinear waves “called solitons” that can maintain its identity even after it collides with another wave with the same kind [1]. This remarkable discovery motivated Russel conduct experiments to study these solitary waves. He empirically eq() below: c^2= g(h+a) ()
The relation () determines the speed of the solitary wave equation, where a is the maximum amplitude above the water surface, h is the finite depth and g is the acceleration of the gravity. Thus, these solitary waves are called the gravity waves.
The topic of solitary waves have inspired many scientist to study more about this phenomena. For instance, the two Dutchmen Korteweg and deVries who derived a nonlinear partial differential equation which is known as KdV equation. The KdV equation has received extensive attention as it is used in modeling the height of a surface of a shallow water in the presence of solitary waves [2].

The simplest form of the KdV wave can be written as follows: ut +auux+uxxx =

There are several approaches by which a soliton and multi-soliton solutions for non-linear equations are obtained. In this research paper, only Adomain Method will be used in solving the KdV equation.
Adomian Decomposition Method (overview)
Adomain decomposition method is commonly used in applied mathematics, and in the area of series solutions in particular. This method was proved to be powerful, effective and can easy to solve linear, nonlinear, and ordinary differential equation, linear and nonlinear integral. The decomposition method demonstrate fast convergence of the solutions.

Adomain Decomposition Method in Solving the KdV
This approach will use Adomain decomposition method to obtain a solitary solution for the KdV equation as follows: ut -6uux+uxxx = 0
And will assume the solution is of the form u(x,0)=-2 (ke^kx)/((1+e^kx )^2