^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

Korteweg de-Vries (K-dV) has wide applications in physics, engineering and fluid mechanics. In this the Korteweg de-Vries equation with traveling solitary waves and numerical estimation of analytic solutions have been studied. We have found some exact traveling wave solutions with relevant physical parameters using new auxiliary equation method introduced by PANG, BIAN and CHAO. We have solved the set of exact traveling wave solution analytically. Some numerical results of time dependent wave solutions have been presented graphically and discussed. This procedure has a potential to be used in more complex system of many types of K-dV equation.

Yu. N. Zaiko studied the presence of a singularity results in that the velocity of long wave perturbations in the system becomes imaginary, which corresponds to the wave propagation in the range of nontransparency [^{rd}-order K-dV equation. We studied the governing two-dimensional 3^{rd} order K-dV equation. After some suitable transformation we got a simple form of K-dV Equation (8). Using a new auxiliary equation method we got the ten sets of travelling wave solution of K-dV equation for real case. There are three cases to be arisen in our study. Two of them are real sense and the other is imaginary concept. After getting the analytical solution of K-dV equation we discussed the ten sets of traveling wave solution numerically and their physical phenomena are described in result and discussion section.

The remarkable form of Korteweg-de Vries nonlinear partial differentiable equation [

which was first introduced by Dutch mathematics Diederik Korteweg and Gustav de Vries in 1895, to describe long water waves in a channel of depth, where

is a constant for fairly long waves,

, u is displacement of wave and g is the acceleration due to gravity. In this section we introduce the method of finding the analytic wave solution to nonlinear evolution equation due to Jing Pang, Chun-quan Bian, and Lu Chao [

This method mainly consists of four steps:

Take the complex solutions of (1) in the form

where u is a real constant. Under the transformation (2), (1) becomes an ordinary differential equation

Take the solutions of (3) in the more general form:

where a_{m} and b_{m} are not zero at the same time, and a_{0}, a_{i} and b_{i} are constants to be determined later. The integer m in (4) can be determined by balancing the highest order nonlinear terms and the highest order linear terms of in (3). satisfies the second-order linear ordinary differential equation

where and are constants for the general solution of (5) as follows:

When

When

When

Note: Let a_{i} = 0,. Equation (4) changes to

The form of (6) has been used in (Deng, Z. G., et al., 2009). If we set b_{i} = 0, (4) changes to

Substitute (4) into (3) and collect all terms with the same order of together. The left-hand side of (3) is converted into a polynomial in. Then, let each coefficient of this polynomial to be zero to derive a set of over-determined partial differential equations for

Solve the algebraic equations obtained in step 3 with the aid of a computer algebra system (such as Mathematica or Maple) to determine these constants. Moreover, the solutions of (5) are well known. Then substituting, , , and the solutions of (5) into (4), we can obtain the exact analytical/traveling wave solutions of (1).

Consider the K-dV equation

describe the evolution of long wave (with large length and measurable amplitude) down a canal with a rectangular cross section. Here u represents the wave amplitude, and represents the vertical velocity of the wave at (x, t), describes the rate of change in amplitude with respect to x and is a dispersion term. This means that if u is the amplitude of wave at some point in space, then u_{x} is the slope of the wave at the point and u_{xx} concavity near the point as given in [9,10]. The existence of solitary waves is due to the balancing effects of and in Equation (8). The nonlinear term in Equation (8) is important because the amplitude of the wave depends on its own rate of change in space; it also represents steepening. The term implies dispersion of different frequency components.

Now we choose the traveling wave transformation (2) i.e. where v = constant.

Substituting these into (8), integrating it with respect to once, and letting the integrating constant to be zero, we have

According to step 2 we get m = 2. Therefore we can write the solution of (9) in the form

That is

where a_{2} and b_{2} are not zero at he same time. By using (5) and from (10), we have

Substituting (10) and (11) into (9) and collecting the coefficients of, and letting it be zero without loss of generality we obtain the system:

From (ix) we get either b_{2} = 0 or b_{2} = −12 and from (v) either a_{2} = 0 or. So, there are three cases to be arisen. For b_{2} = 0 and a_{2} = 0 uses the system of equations (i) to (ix) we get trivial solutions.

Trivial solution set is a_{0} = a_{1} = a_{2} = b_{1} = b_{2} = 0 and the other solution sets are as follows:

For and b_{2} = 0, using the system of Equations (i)-(ix) we get a set of solution is as follows:

For a_{2} = 0 and b_{2} = −12, using the system of Equations (i)-(ix) we get a set of solutions are as follows:

For and b_{2} = −12, using the system of Equations (i)-(ix) we get a set of solutions are as follows:

and

where and are arbitrary constants. By using (A)- (E) Equation (10) can be written as:

Equations (A)-(E) and (10) implies respectively as follows:

Now, the second order differential Equation (5) is as follows:

When,

When

When

For the first case:

where,

Therefore, Equations (F)-(J) becomes:

where

where

We study here the interaction of two wave solutions to the third order K-dV equation. The numerical representation of two dimensional third order K-dV equations for this problem are obtained by the Fortran scheme and compared with analytical solution cases for, , and c = 10.0 such that. Numerical representation generates that the same behavior as wave solutions. The solutions remain unchanged before and after their interaction. As seen in

example of two solution on the domain [0, 40] × [0, 1]. As expected, we see that the soliton travels until they collide, but their amplitudes are unchanged and position in time has changed considerably. Amplitude of u wave with respect to t and x gradually increases smoothly. In

x = 15.0, x = 20.0 and x = 25.0.

this figure we see that the behavior of the u wave is exactly what we expect in the analytical sense, the only thing that the change about the wave once in time for x = 5.0, x = 10.0, x = 15.0, x = 20.0 and x = 25.0 anywhere else about the wave stays the same. In

For the second case:

,

Therefore,

Equations (F)-(J) and (10) implies respectively as follows:

which is same as Equation (20).

The numerical representation of two dimensional third order K-dV equation for this problem is obtained by the Fortran scheme compared with analytical solution to the following cases for, and c = 12.0 such that in the real sense. Numerical solution generates that the same behavior as solitary wave solutions for different cases. The solutions remain unchanged before and after their intersection. As seen in Figures 11(a) and (a1) for the Equation (17), time evaluation of u wave for different values of displacement like x = 5.0, x = 10.0, x = 15.0, x = 20.0 and x = 25.0. For a particular value of x = 10.0, u wave simply solitary. But the second case Figures 11(b) and (b1) for the same equation, u-wave generates solitary for different values of t = −0.4, t = 0.0, t = 0.4, t = 0.8 and t = 1.2. Here

negative sign of t indicates before the collision. For a particular value of t = 0.0 the u wave is solitary. In

= 12.0, c = 15.0, c = 17.0. But in

In this research numerical estimation of traveling wave solution for third order of two-dimensional K-dV equation using a new auxiliary equation method has been studied. The K-dV equation for the present problem comes from the third order two dimensional governing Equation (1.1) after some suitable transformation. It is found that there are nine exact traveling wave solutions (12)-(20) of 2-dimentional K-dV equation exist for real sense depends on different relevant physical parameters but the last one is exact the same as (20). Numerical results of first ten cases for real sense obtained by using FORTRAN program have been shown graphically and discussed. We have also found that when employing the Fortran-Scheme for the numerical estimation of K-dV equation that are presented graphically for the first case λ^{2} – 4μ > 0 and second case λ^{2} – 4μ > 0 Equations (12)- (20). Remaining imaginary cases will be avenue of another research work in the future.