Hyperbolic , Trigonometric , and Rational Function Solutions of Hirota-Ramani Equation via G ′ / G-Expansion Method

The G′/G -expansion method is proposed to construct the exact traveling solutions to HirotaRamani equation: ut −uxxt aux 1−ut 0, where a/ 0. Our work is motivated by the fact that the G′/G -expansionmethod provides not only more general forms of solutions but also periodic and solitarywaves. If we set the parameters in the obtainedwider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.


Introduction
Nonlinear evolution equations NLEEs have been the subject of study in various branches of mathematical-physical sciences such as physics, biology, and chemistry.The analytical solutions of such equations are of fundamental importance since a lot of mathematicalphysical models are described by NLEEs.Among the possible solutions to NLEEs, certain special form solutions may depend only on a single combination of variables such as solitons.In mathematics and physics, a soliton is a self-reinforcing solitary wave, a wave packet or pulse, that maintains its shape while it travels at constant speed.Solitons are caused by a cancelation of nonlinear and dispersive effects in the medium.The term "dispersive effects" refers to a property of certain systems where the speed of the waves varies according to frequency.Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.The soliton phenomenon was first described by John Scott Russell 1808-1882 who observed a solitary wave in the Union Canal in Scotland.He reproduced the phenomenon in a wave tank and named it the "Wave of Translation" 1 .Many exactly solvable models have soliton solutions, including the Korteweg-de Vries equation, the nonlinear Schr ödinger equation, the coupled nonlinear Schr ödinger equation, the sine-Gordon equation, and Gardner equation.The soliton solutions are typically obtained by means of the inverse scattering transform 2 and owe their stability to the integrability of the field equations.In the past years, many other powerful and direct methods have been developed to find special solutions of nonlinear evolution equations NEE s , such as the Bcklund transformation 3 , Hirota bilinear method 4 , numerical methods 5 , and the Wronskian determinant technique 6 .With the help of the computer software, many algebraic methods are proposed, such as tanh method 7 , F-expanded method 8 , homogeneous balance method 9 , Jacobi elliptic function method 10 , the Miura transformation 11 , and some other new methods 12, 13 .Recently, the G /G -expansion method, firstly introduced by Wang et al. 14 , has become widely used to search for various exact solutions of NLEEs 14-18 .The value of the G /G -expansion method is that one treats nonlinear problems by essentially linear methods.The method is based on the explicit linearization of NLEEs for traveling waves with a certain substitution which leads to a second-order differential equation with constant coefficients.Moreover, it transforms a nonlinear equation to a simple algebraic computation.
Our first interest in the present work is in implementing the G /G -expansion method to stress its power in handling nonlinear equations, so that one can apply it to models of various types of nonlinearity.The next interest is in the determination of new exact traveling wave solutions for the Hirota-Ramani equation 11-13 : where a / 0 is a real constant and u x, t is the amplitude of the relevant wave mode.This equation was first introduced by Hirota and Ramani in 11 .Ji obtained some travelling soliton solutions of this equation by using Exp-function method 13 .This equation is completely integrable by the inverse scattering method.Equation 1.1 is studied in 11-13 where new kind of solutions were obtained.Hirota-Ramani equation is widely used in various branches of physics, and such as plasma physics, fluid physics, and quantum field theory.It also describes a variety of wave phenomena in plasma and solid state 11 .

Description of the G /G -Expansion Method
The objective of this section is to outline the use of the G /G -expansion method for solving certain nonlinear partial differential equations PDEs .Suppose we have a nonlinear PDE for u x, t , in the form where P is a polynomial in its arguments, which includes nonlinear terms and the highestorder derivatives.The transformation u x, t U ξ , ξ kx ωt, reduces 2.1 to the ordinary differential equation ODE where U U ξ , and prime denotes derivative with respect to ξ.We assume that the solution of 2.2 can be expressed by a polynomial in G /G as follows: where α 0 , and α i , are constants to be determined later, G ξ satisfies a second-order linear ordinary differential equation LODE : where λ and μ are arbitrary constants.Using the general solutions of 2.4 , we have and it follows, from 2.3 and 2.4 , that and so on, here the prime denotes the derivative with respect to ξ.
To determine u explicitly, we take the following four steps.
Step 1. Determine the integer m by substituting 2.3 along with 2.4 into 2.2 , and balancing the highest-order nonlinear term s and the highest-order partial derivative.
Step 2. Substitute 2.3 , given the value of m determined in Step 1, along with 2.4 into 2.2 and collect all terms with the same order of G /G together, the left-hand side of 2.2 is converted into a polynomial in G /G .Then set each coefficient of this polynomial to zero to derive a set of algebraic equations for k, ω, λ, μ, α 0 and α i for i 1, 2, . . ., m.
Step 3. Solve the system of algebraic equations obtained in Step 2, for k, ω, λ, μ, α 0 and α i by use of Maple.
Step 4. Use the results obtained in above steps to derive a series of fundamental solutions u ξ of 2.2 depending on G /G ; since the solutions of 2.4 have been well known for us, then we can obtain exact solutions of 2.1 .

Application on Hirota-Ramani Equation
In this section, we will use our method to find solutions to Hirota-Ramani equation 10-12 : where a / 0. We would like to use our method to obtain more general exact solutions of 3.1 by assuming the solution in the following frame: where k, ω are constants.We substitute 3.2 into 3.1 to obtain nonlinear ordinary differential equation By setting U V, nonlinear ordinary differential equation 3.3 reduce to ω ak V − k 2 ωV − akωV 2 0.

According to
Step 1, we get m 2 2m, hence m 2. We then suppose that 3.4 has the following formal solutions: where α 2 , α 1 , and α 0 , are unknown to be determined later.
Substituting 3.5 into 3.4 and collecting all terms with the same order of G /G , together, the left-hand sides of 3.4 are converted into a polynomial in G /G .Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for λ, μ, α 0 , α 1 , and α 2 , as follows: and solving by use of Maple, we get the following results.

Case 1.
λ − 1 6 where k, ω, and α 1 are arbitrary constants.Therefore, substitute the above case in 3.5 , and using the relationship U ξ V ξ dξ, we get Substituting the general solutions 2.5 into 3.8 , we obtain three types of traveling wave solutions of 3.1 in view of the positive, negative, or zero of λ 2 − 4μ.When D 1 λ 2 − 4μ ak ω /k 2 ω > 0, using the integration relationship 3.8 , we obtain hyperbolic function solution U H , of Hirota-Ramani equation 3.1 as follows:

Rational Solution
And finally, in both Cases 1 and 2, when D λ 2 − 4μ 0, we obtain rational solution:

Conclusions
This study shows that the G /G -expansion method is quite efficient and practically well suited for use in finding exact solutions for the Hirota-Ramani equation.Our solutions are in more general forms, and many known solutions to these equations are only special cases of them.With the aid of Maple, we have assured the correctness of the obtained solutions by putting them back into the original equation.We hope that they will be useful for further studies in applied sciences.
16respectively, where ξ kx ωt, and C 1 , C 2 , are arbitrary constants.The trigonometric function solution 3.16 , for a −1, k −1, ω 1/2, C 1 2 and C 2 1 are shown in Figure3.Similarly, to obtain some special forms of the solutions obtained above, we set C 2