Solitary Wave Solution

Solitary Wave Solution has been developed, for regularized long width equation in which the auxiliary equation different from traditional (G`/G)-Expansion Method. This method is more general and very simple. This method is more reliable and efficient.

4.1 Introduction

John Scott Russell was the first one who observed the solitary waves in 1834. He observed a large protrusion of water slowly travelling on the Edinburgh-Glasgow canal without changing its shape. He observed the bulge of water and called it ‘‘great wave of translation” [1] was travelling along the channel of water for a long period of time while still retaining its shape. The single humped wave of bulge of water is now called solitary wave or soliton. In 1895, Diederik Korteweg and Gustav de Vries provided what is now known as the Korteweg de Vries equation (KdV), they also gave its solitary wave and periodic conidial wave Solutions. In 1965, Norman Zabusky and Martin Kruskal investigated numerically the nonlinear interaction of a large solitary-wave with a smaller one, and the recurrence of initial states [2]. They discovered that solitary waves undergo nonlinear interaction following with KdV equation. The remarkable discovery, of Russell that solitary waves retain their identities and their character resembles particle like behavior, motivated Zabusky and Kruskal to call these solitary waves to solitons.

We will discusses the traveling wave Solutions of nonlinear evolution equations tackled by new approach of (G`/G)-expansion method on Regularized Long Width (RLW) Equation, Modified Equal Width Equation, Burger Hierarchy Equation, Symmetric Regularized Long Wave Equation, Hirota Ramani Equation, Nonlinear Boussnique Equation. The physical properties of several nonlinear traveling wave Solutions by plotting and analyzing their figures have also been studied.

4.2 New Approach of (G`/G)-Expansion Method

Recently a new approach of (G`/G)-Expansion Method has been developed, in which the auxiliary equation different from traditional (G`/G)-Expansion Method. This method is more general and very simple. The number of Solutions more than (G`/G)-Expansion Method.

4.1.1 Methodology

We consider the general nonlinear PDE of the type

    (4.1)

where  P is a polynomial in its arguments. The essence of the (G`/G)-expansion method can be presented in the following steps:

Step 1: Seek Solitary wave Solutions of Eq. (4.1) by taking

and transform Eq. (4.1) to the ordinary differential equation.

    (4.2)

where   and m is constant and where prime denotes the derivative with respect to .

Step 2: If possible, integrate Eq. (4.2) term by term one or more times. This yields constant(s) of integration. For simplicity, the integration constant(s) can be set to zero.

Step 3: According to (G`/G)-expansion method, we assume that the wave solution can be expressed in the following form

    (4.3)

where is the solution of first order nonlinear equation in the form

    (4.4)

            .

            .

            ..

where   and are real constants. The Riccati Eq. (4.4) plays important role in manipulating nonlinear equations to get exact Solutions by the (G`/G)-expansion method. It has the following type of exact Solutions.

Family 1: When , ,

Family 2: When

Family 3: When

Family 4: When

Family 5: When

Step 4: Determine M. This, usually, can be accomplished by balancing the linear term(s) of highest order with the highest order nonlinear term(s) in Eq. (4.2).

Step 5: Substituting (4.3) into Eq. (4.2) with (4.4) will yields an algebraic equation involving power of (G`/G). Equating the coefficients of like power of (G`/G) to zero gives a system of algebraic equations for a. Then, we solve the system with the aid of a computer algebra system (CAS), 5such as MAPLE 13, to determine these constants.

Step 6: Putting these constant into Eq. (4.4), coupled with the well known Solutions of Eq. (4.4), we can obtained the more general type and new exact traveling wave solution of the nonlinear partial differential equation (4.1).

4.3 Applications

4.3.1  Regularized Long Width Equation

Consider the Regularized Long Width equation [3]

    (4.5)

To convert Eq. (4.5) into ODE we use following transformation

    (4.6)

where k  and are arbitrary constant. Substituting Eq. (4.6) into Eq. (4.5) and using the chain rule and  we obtained we obtained

   (4.7)

Consider the trial solution for Eq. (4.5) as follow

    (4.8)

where  can be obtained by applying the homogenous balancing principle we have

    (4.9)

Using the value of into Eq. (4.8), we obtained the trail solution

     (4.10)

where satisfying the following Riccati equation

Putting Eq. (4.8) into Eq. (4.7) coupled with Eq. (4.11); the Eq. (4.7) yields an algebraic equation involving power of   as

Compare the like powers of we have system of equations

Solving the above system for unknown parameters, we have the following solution sets

  Solution Set

Family 1: When

where

Family 2: When

where

Family 3: When

where

Family 4: When

where

Family 5: When

where

In all cases

 

                                               Fig 4.1 Solitary Wave Solution for the given PDE (4.5)

  Fig 4.1 (b) 3D Solitary Wave Solution of Eq. (4.5)  for Different Values of Parameters

Fig 4.1 (a) 2D Solitary Wave solution of Eq. (4.5) for Different Values of Parameters

Solution Set

Family 1: When

where

Family 2: When

where

Family 3:When

where

Family 4: When

where

Family 5: When

where

In all cases

                                      Fig 4.1 Solitary Wave Solution for the given PDE (4.5)

Fig 4.1 (d) 3D Solitary Wave solution of  for  Different Values of Parameters

Fig 4.1 (c) 2D Solitary Wave Solution of Eq. (4.5)  for Different values of Parameters

References

[1]. J. S. Russell, Report on Waves: 14th Meeting of the British Association for the Advancement of Science. John Murray, London. 311–390. (1844).

[2]. N.J. Zabusky and M. D. Kruskal, Interaction of solitons in collisionless plasma and the recurrence of initial states. Physical Review Letters. 15 (6), 240–243, (1965).

[3]. A.M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, HEP and Springer, Beijing and Berlin. (2009).