Periodic, kink and bell shape wave solutions to the Caudrey-Dodd-Gibbon (CDG) equation and the Lax equation

This paper addresses the implementation of the new generalized (G′/G)expansion method to the CaudreyDodd-Gibbon (CDG) equation and the Lax equation which are two special case of the fth order KdV (fKdV) equation. The method works well to derive a new variety of travelling wave solutions with distinct physical structures such as soliton, singular soliton, kink, singular kink, bell-shaped soltion, anti-bell-shaped soliton, periodic, exact periodic and bell type solitary wave solutions. Solutions provided by this method are numerous comparing to other methods. To understand the physical aspects and importance of the method, solutions have been graphically simulated. Our results unquestionably disclose that new generalized (G′/G)expansion method is incredibly in uential mathematical tool to work out new solutions of various types of nonlinear partial di erential equations arises in the elds of applied sciences and engineering.


Introduction
It is well observed that almost every natural phenomena is nonlinear and mathematically which appears in the form of nonlinear evolution equations (NLEEs).The studies of NLEEs, a special type of nonlinear partial dierential equations (NPDEs), becomes one of the most exciting and extremely active areas of research and investigation because several problems in various scientic and engineering elds, such as solid state physics, chemical physics, plasma physics, optics, biology, chemical kinematics, geochemistry, uid mechanics and hydrodynamics are frequently describe by NLEEs. To understand the internal mechanism of these problems, nding the exact traveling wave solutions is becoming more and more fascinating day-by-day in nonlinear science. But there is not any integrated method which could be utilized to deal with all types of NLEEs. That is why a variety of ecient and reliable methods have been developed. For example, the Painleve expansion method [1], the inverse scattering method [2,3], the Darboux transformation method [4,5], the Cole-Hopf transformation method [6,7] the Jacobi elliptic function method [8,9], the Hirotaâs bilinear transformation method [10,11], the Backlund transformation method [12,13], the sineâcosine function method [14], the tanh method [15][16][17], the improved F-expansion method [18], the tanh-coth method [19], the exp-function method [20], the exp the exp (−φ(ξ))-expansion method [21], the modied simple equation method [22,23], the (G /G)-expansion method [24,25], the novel (G /G)-expansion method [26], the improved (G /G)expansion method [27], the generalized (G /G)-expansion method [28,29], the double (G /G, 1/G)-expansion method [ 30] , the modied sineâcosine function method [31], the canonical transformation method [32], the compatible transform method [33], the new generalized φ 6 -model expansion method [34], the homotopy analysis method [35] and so on.

the integrable nonlinear fth order Lax equation is of the form [32]
u t + u xxxxx + 10uu xxx + 20u x u xx + 30u 2 u x = 0.
In this paper we aim is to investigate Eq. (1) and Eq. (2) using the generalized (G /G)-expansion method to explore more exact solutions which include new periodic, soliton and kink solutions. This paper is organized as follows: In Section 2, we will review briey the generalized (G /G)-expansion method.In Section 3, we present the application of the methods to Eq. (1) and Eq. (2)and the obtained solutions. In Section 4, we give the physical and graphical presentation of the obtained results. Finally, In Section 5, conclusions are drawn.

Description of the New Generalized (G /G)-expansion Method
Let us consider a general nonlinear PDE in the form where u = u(x, t) is an unknown function, P is a polynomial in u(x, t) and its partial derivatives in which the highest order derivatives and the nonlinear terms are involved.The main steps of the generalized (G /G)expansion method are as follows: Step 1: We suppose that the combination of real variables x and t by a variable ξ as follows: where c is the speed of the traveling wave. The traveling wave transformation (4)allows us to reduce equation (3) to an ODE for u=u(ξ) in the form where R is a function of u(ξ)and the superscripts indicate the ordinary derivatives withrespect to ξ.
Step 2: In many instances, equation (5) can be integrated term by term one or more times, yielding constants of integration, which can be set equal to zero for straightforwardness.
Step 3: We assume that the traveling wave solution of equation (5) can be expressed as follows: where either a N or b N may be zero, but both a N and b N cannot be zero at a time, a k (k = 0, 1, 2, 3, . . . N ), b k (k = 1, 2, 3, . . . N )and d are arbitrary constants to be determined later and Y (ξ) is given by where G = G(ξ) satises the following auxiliary nonlinear ordinary dierential equation where prime indicates the derivative with respect to ξ and A, BC, E are real parameters.
Step 4: The positive integer N can be determined by using the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (5).
Step 6: We stare that the values of the constants can be determined by solving the algebraic equations achieved in Step 5. Since the general solution of (7) is in general known, inserting the value of a k (k= 0, 1, 2, . . ., N ), b k (k= 1, 2, . . ., N ), d and c into (6) yields the comprehensive and newly produced exact traveling wave solutions of the nonlinear partial dierential equation (3).
Step 7: By using the general solution of equation (8), we admit the following solution of equation (7).

Applications
In this section, the new generalized (G /G)-expansion method has been put in use to examine travelling wave solutions of the nonlinear fth order Caudrey-Dodd-Gibbon (CDG) and Lax equations.
According to the method described in Section 2, and after balancing we obtain N = 2. Therefore, we seek solutions to (16) in the form where a 0 , a 1 , a 2 , b 1 , b 2 and d are arbitrary constants to be determined later. Now substituting Eq.(17) into Eq. (16) and using (7) and (8), the left hand side of Eq. (16) is translated into the polynomials in (d+Y ) N , (N = 0, 1, 2, . . .) and (d+Y ) −N , (N = 1, 2, . . .). Equating the coecients of these polynomials to zero, we obtain an algebraic system (for simplicity, we leave out the displaying of the equations) with respect to a 0 , a 1 , a 2 , b 1 , b 2 , c and d.
Solving the system of algebraic equations with the aid of the Maple 17 yields the following families of values of a 0 , a 1 , a 2 , b 1 , b 2 , c and d.
For Case 1: From Case 1 putting the values of constants into Eq. (17) and combining with Eqs. (9) to (12) and simplifying, we attain following traveling wave solutions for r = 0 but s = 0 respectively In similar fashion, substituting the values of the constants arranged in Eq. (18) into Eq. (17), as well as (9) to (12) and simplifying, we attain following traveling wave solutions for s = 0 but r = 0 respectively But in case of Eq. (13) we didn't admit any kind (for r = 0 but s = 0 and s = 0 but r = 0) travelling wave solution because in this case travelling wave velocity became zero.
For Case 2: Proceeding as before, making use of the values of constants in Case 2 into Eq. (17) along with Eqs. (9) to (12) we have the following traveling wave solutions for r = 0 but s = 0 respectively Now, inserting (19) into (17) and using (9) to (12), respectively we get the traveling wave solutions as for s = 0 but r = 0 For Family 5 traveling wave solutions are not admit able because obtained velocity of the wave is zero.
For Case 3: Also, from Case 3 placing the values of constants provided in Eq. (20) into Eq. (17) accompanied with (9) to (12) and after simplication, respectively we nd the following travelling solutions for r = 0 but s = 0 Furthermore, substituting (20) into Eq. (17) along with (9) to (12) and simplifying, respectively we nd the following travelling solutions for s = 0 but r = 0 Furthermore, when we combine with Eq. (13) with Case 3 we nd no traveling wave speed and that's why it became impossible to get any traveling wave solutions for Eq. (13).
Similarly, Case 3, Case 4 and Case 5 exert traveling wave solutions of CDG equation for shake of simplicity which aren't reported here.
Example 3.2. In this section we will examine the nonlinear fth order Lax equation (2).
The fth order Lax equation (2) can be rewritten as Applying ξ=x−ct, equation (24) converts into the following ODE for u (x, t) =v(ξ), Integrating (25), setting the constant of integration to zero, we obtain Balancing the highest order linear term v (iv) and nonlinear term of the highest order v 3 in equation (26), yields N = 2. Therefore, the solution of equation (19) appears in the following form: where a 0 , a 1 , a 2 , b 1 , b 2 and d are arbitrary constants to be determined.
Substituting (27) accompanied with (7) and (8) into (26), the left-hand side is diverted into the polynomials in (d+Y ) N , (N = 0, 1, 2, . . .) and (d+Y ) −N , (N = 1, 2, . . .). We draw together each coecient of this resulted polynomial and setting them to zero yields an over determined set of algebraic equations (for simplicity the equations are not presented here) for a 0 , a 1 , a 2 , b 1 , b 2 , c and d. Solving these algebraic equations with the help of symbolic computation software, such as, Maple 17, we obtain the following Case 1: where ψ=A−C,Ω = Eψ, d, A, B, C and E are free parameters.
For Case 1: By use of the values of constants from Case 1 into Eq. (27) and combining with Eqs. (9) to (12) we obtain the following travelling wave solutions for r = 0 but s = 0 respectively Again, substituting the values of the constants arranged in Eq. (28) into Eq. (27), as well as Eqs. (9) to (12) and simplifying, we attain following traveling wave solutions for s = 0 but r = 0 respectively Since wave speed become zero for Eq. (13) when combine with (27) so traveling wave solution is not attainable.
For Case 2: In similar fashion, determined values of the constants, presenting in Case 2, putting into (27) accompanied with (9) to (12) respectively we obtain the travelling wave solutions for r = 0 but s = 0 as follows Again setting (29) into Eq. (27) along with (9) to (12) and simplifying we get following traveling wave solutions for s = 0 but r = 0 respectively Also using (29) in to (27) together with (13) no traveling wave solution exist for both cases (when r = 0 but s = 0 and s = 0 but r = 0).

For Case 3:
By means of the values of the constants contained in Eq. (30) into (27), together with Eqs. (9) to (12) and simplifying, we attain the traveling wave solutions as follows for r = 0 but s = 0 respectively Again, substituting (30) into (27), accompanied with Eqs. (9) to (12) and after simplication, we nd the following traveling wave solutions for s = 0 but r = 0 respectively In this case also traveling wave solution for (13) is not exist able.

Conclusion
In this research, we succeeded in applying generalized (G /G) expansion method on two specic fth order KdV (fKdV) equations namely, CDG equation (1) and Lax equation (2). And we successfully obtained wider classes of exact travelling wave solutions with a variety of distinct physical structures such as soliton, singular soliton, kink, singular kink, bell-shaped soltion, anti-bell-shaped soliton, periodic, exact periodic and bell type solitary wave solutions which are shown in Fig. 1-Fig. 5. On comparing our results in this paper with the well-known results obtained in [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57], most of the obtained solutions are exclusively new. The pivotal privilege of this implemented method against other methods is that the method provides more general and huger amount of new wave solutions which validate the superiority of this method.