The Investigation of Exact Solutions for the Appropriate Type of the Dispersive Long Wave Equation

Improved (G 󸀠 /G)-expansion and first integral methods are used to construct exact solutions of the (2 + 1)-dimensional Eckhaustype extension of the dispersive long wave equation.The (G 󸀠 /G)-expansion method is based on the assumptions that the travelling wave solutions can be expressed by a polynomial in (G 󸀠 /G) and the first integral method is based on the theory of commutative algebra in which Division Theorem is of concern. It is worth mentioning that these methods are used for different systems and those two different systems can both be reduced to a system that will be mentioned in this paper. To recapitulate, this investigation has resulted in the exact solutions of the given systems.


Introduction
The investigation of exact solutions to nonlinear evolution has become an interesting subject in nonlinear science field, since the time when the soliton concept was first introduced by Zabusky and Kruskal in 1965 [1].It was not until the mid-1960s when applied scientists began to use modern digital computers to study nonlinear wave propagation that the soundness of Russell's early ideas began to be appreciated.He viewed the solitary wave as a self-sufficient dynamic entity; a "thing" displaying many properties of a particle.From the modern perspective it is used as a constructive element to formulate the complex dynamical behavior of wave systems throughout science: from hydrodynamics to nonlinear optics, from plasmas to shock waves, and from the elementary particles of matter to the elementary particles of thought.For a more detailed and technical account of the solitary wave, see [2].

Methodology
Consider a general nonlinear partial differential equation in the form where  = (, , ) is the solution of nonlinear PDE Equation (4).Furthermore, the transformations which are used are as follows: where , , and  are constants.Using the chain rule, it can be found that At present, ( 6) is employed to change the nonlinear PDE Equation (4) to nonlinear ordinary differential equation  ( () ,   () ,   () , . ..) = 0.
The general solutions of the second order LODE (9) have been well known for where  1 and  2 are arbitrary constants.

The First Integral Method
. By focusing on (4), a new independent variable is introduced as This yields a system of nonlinear ODEs If it is revealed that the integrals to (14) are under the same conditions of the qualitative theory of ordinary differential equation [17], then general solutions to ( 14) can be solved directly.However, it is generally so difficult for us to realize this even for one first integral.That is why for a given plane autonomous system there is no systematic theory that can tell us how to find its first integrals, nor is there a logical way for telling us what these first integrals are.Thus, Division Theorem is used to obtain one first integral of (14).Now, let us recall the Division Theorem.

Application
In this paper, improved (  /)-expansion method is employed to study the solutions of the nonlinear partial differential system (2); afterwards, the first integral method is employed to study the solutions of the nonlinear partial differential system (3).

Explicit and Exact Solution of the System (2)
. We introduce the transformations, Substituting ( 15) into ( 2), there will be a change as follows: where by twice integrating the first equation of ( 16), in respect to , it can be found that By integrating the second equation of ( 16), in respect to , then substituting (17) into this equation, it can be found that where  1 and  2 are arbitrary integration constants that are to be determined later.Suppose that the solution of ODE ( 18) can be expressed by a polynomial in (  /) as follows: where  = () satisfies the second order LODE (9).By balancing the term   with the term  3 () in (18), we obtain  = 1, so we can write (19) as And therefore By substituting (20) and (21) into ODE (18) and collecting all terms with the same power of (  /) together, the lefthand side of ODE ( 18) is converted into another polynomial in (  /).Equating each coefficient of this polynomial to zero yields a set of simultaneous algebraic equations for  0 ,  1 ,  1 ,  1 ,  2 , , and .Having solved the given equation with aid Maple, the following solutions will be attained: By using ( 22)-( 26), expression (20) can be written as follows: Equations ( 27)-(31) are the formula of solution of (18).Substituting the general solutions of ( 9) into ( 27)-(31) we have three types of travelling wave solutions of the (2 + 1)dimensional Eckhaus-type equation (2) (so    is definiendum th type solution from th expression) as follows.
where  1 and  2 are arbitrary constants.

ISRN Mathematical Physics
Case C. When  2 − 4 = 0, where where by once integrating the first equation of (36), with respect to , it can be found that By twice integrating the second equation of ( 16), with respect to  and then substituting (37) into this equation, it can be found that where  1 and  2 are arbitrary integration constants that are to be determined later.