Application of the generalized Kudryashov method to the Eckhaus equation

It is well known that nonlinear evolution equations (NLEEs) are widely used to describe physical phenomena in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, etc. [1–7, 10, 12–15, 20, 21, 23–29, 31–36]. In order to understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties. Solutions for the NLEEs can not only describe the designated problems, but also give more insights on the physical aspects of the problems in the related fields. In recent years, various powerful methods have been presented for finding exact solutions of the NLEEs in mathematical physics, such as tanh-function method [28], extended tanh-function method [10,29], sine-cosine method [27], Jacobi elliptic function method [13, 32], F -expansion method [1, 31], exp-function method [14], (G′/G)-expansion method [26], Q-function method [20] and so on. The Q-function method, which is a direct and effective algebraic method for computing exact travelling wave solutions, was first proposed by Kudryashov [16]. The Q-function method that is known as the Kudryashov method is one of the most effective methods


Introduction
It is well known that nonlinear evolution equations (NLEEs) are widely used to describe physical phenomena in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, etc. [1-7, 10, 12-15, 20, 21, 23-29, 31-36].In order to understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties.Solutions for the NLEEs can not only describe the designated problems, but also give more insights on the physical aspects of the problems in the related fields.In recent years, various powerful methods have been presented for finding exact solutions of the NLEEs in mathematical physics, such as tanh-function method [28], extended tanh-function method [10,29], sine-cosine method [27], Jacobi elliptic function method [13,32], F -expansion method [1,31], exp-function method [14], (G /G)-expansion method [26], Q-function method [20] and so on.
The Q-function method, which is a direct and effective algebraic method for computing exact travelling wave solutions, was first proposed by Kudryashov [16].The Q-function method that is known as the Kudryashov method is one of the most effective methods

The generalized Kudryashov method
Suppose that we have a nonlinear evolution equation in the form where u = u(x, t) is an unknown function, F is a polynomial in u and its various partial derivatives u t , u x with respect to t, x respectively, in which the highest order derivatives and nonlinear terms are involved.
Step 1.Using the traveling wave transformation where ξ 0 is an arbitrary constant and k, c are constant to be determined later.Then Eq. ( 1) is reduced to a nonlinear ordinary differential equation (NODE) of the form Step 2. Suppose that the solution of Eq. ( 3) has the following form: where a i (i = 0, 1, . . ., N ) and b j (j = 0, 1, . . ., M ) are constants to be determined such that a N = 0, b M = 0 and http://www.mii.lt/NA is the solution of the equation where κ is an arbitrary constant.
Step 3. Determine the positive integer numbers N and M in Eq. ( 4) by using the homogeneous balance between the highest order derivatives and the nonlinear terms in Eq. ( 3) after substituting Eq. ( 6) and the necessary derivatives of u, which have the form where the prime denotes the derivative d/dQ.
Step 4. Substitute Eqs. ( 4), ( 7) and ( 8) into Eq.( 3).As a result of this substitution, we get a polynomial of Q.In this polynomial we gather all terms of same powers and equating them to be zero, we obtain a system of algebraic equations, which can be solved by the Maple or Mathematica to get the unknown parameters a i (i = 0, 1, . . ., N ), b j (j = 0, 1, . . ., M ), k, c.Consequently, we obtain the exact solutions of Eq. ( 1).

The Eckhaus equation
The Eckhaus equation is a nonlinear Schrödinger-type equation, which can be written as This equation has been solved by using the (G /G)-expansion method [30].Let us now solve Eq. ( 9) by using the generalized Kudryashov method.To this end, we use the following wave transformation: where k, α, β are constants to be determined later and ξ 0 is an arbitrary constant.Now Eq. ( 9) is reduced to the following NODE: Then Eq. ( 11) can be reduced to the following NODE: Balancing vv ξξ with v 4 in Eq. ( 13), we get the formula Substituting v and it's their necessary derivatives into (13) and equating all the coefficients of Q to zero, we obtain http://www.mii.lt/NASolving system ( 15)-( 23) with the aid of Mathematica, we obtain the following results.

Conclusions
In this paper, we have proposed the generalized Kudryashov method for solving the Enkhaus equation.This work has illustrated that the solutions obtained in [30] are considered as a special case of our obtained solutions and a new results have been obtained using this method.This method is direct, effective and can be extended for solving many systems of nonlinear PDEs.The soliton solutions that are retrievable from this equation are topological and singular soliton solutions only.Being a dissipative model, it is not possible to obtain non-topological soliton solution.Therefore, it makes sense that this model only retrieves singular and topological soliton solutions.