Exact travelling wave solutions of the symmetric regularized long wave (SRLW) using analytical methods

In this article, we establish exact travelling wave solutions of the symmetric regularized long wave (SRLW) by using analytical methods. The analytical methods are: the tanh-coth method and the sech method which used to construct solitary wave solutions of nonlinear evolution equations. With the help of symbolic computation, we show that aforementioned methods provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.


Introduction
Recently, the investigation of exact travelling wave solutions to nonlinear partial differential equations plays an important role in the study of nonlinear modelling physical phenomena.Also the study of the travelling wave solutions plays an important role in nonlinear science.Nonlinear evolution equations are widely used as models to describe complex physical phenomena and have a significant role in which arises in several physical applications including ion sound waves in plasma [18].The article is organized as follows: In Sections 2 and 3 first we briefly give the steps of the methods and apply the methods to solve the nonlinear partial differential equations.In Section 4, the application of the analytical methods to the symmetric regularized long wave (SRLW) will be introduced briefly.Also a conclusion is given in Section 5. Finally some references are given at the end of this paper.

Basic idea of tanh-coth method
The standard tanh method is well-known analytical method which first presented by Malfliet's [21] and developed in [21,22].By summarizing tanh-coth method by Wazwaz [23] for a given NLPDE with independent variables X = (x, y, z, t) and dependent variable u: can be converted to on ODE which transformation ξ = k 1 x + k 2 y − ct is wave variable.Also, c, k 1 and k 2 are constants to be determined later.Introducing a new independent variable leads to the change of derivatives The tanh-coth method [24] admits the use of the finite expansion where a k (k = 0, 2, ..., m), b k (k = 1, 2, ..., m) and µ are constants to be determined later, but the degree of which is generally equal to or less than m − 1, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. ( 3).If m is not an integer, then a transformation formula should be used to overcome this difficulty.For aforementioned method, expansion (7) reduces to the standard tanh method [21] for b k = 0, 1 ≤ k ≤ m.Substituting Eq. ( 7) into the ODE results is an algebraic system of equations in the powers of Y that will lead to the determination of the parameters a k (k = 0, 2, ..., m), b k (k = 1, 2, ..., m) and c.To show the efficiency of the method described in the previous part, we present some examples.

Basic idea of sech 2 method
We now describe the sech 2 method for the given partial differential equations.We give the detailed description of method which to use this method, we take following steps: Step 1.For a given NLPDE with independent variables X = (x, y, z, t) and dependent variable u, we consider a general form of nonlinear equation: which can be converted to on ODE which transformation ξ = k 1 x + k 2 y − ct is wave variable.Also, c, k 1 and k 2 are constants to be determined later.
Step 2. We introduce a new independent variable as following leads to the change of derivatives in the form where other derivatives can be derived in a similar manner.If we use a new independent variable: leads to the change of derivatives in the form The sech 2 method admits the use of a finite expansion of sech function where a 0 , a k (k = 1, 2, ..., m) and µ are constants to be determined later.but, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. (10).
SYMMETRIC REGULARIZED LONG WAVE 51 and using the wave variable ξ = k 1 x − ct reduce it to an ODE Integrating Eq. ( 21) with respect to ξ and considering the zero constants for integration, we obtain Balancing the terms that involve u ′′′ and uu ′ in Eq. ( 22) gives The tanh-coth method allows us to use the substitution Substituting Eq. ( 25) in to Eq. ( 22), with the help of Maple gives the following set of non-trivial solutions or or where c and µ are arbitrary constants.Substituting Eqs. ( 26)-(28) into expression Eq. ( 25), can be written as or or It is worth to point out that the following periodic solutions or or which are the exact solutions of symmetric regularized long wave (SRLW) equation.Can be seen that the results are the same, with comparing results Darwish's and Xu's [18].

Using the sech 2 method
Considering the following Symmetric Regularized Long Wave (SRLW) equation by using the sech 2 method, and proceeding as before we obtain and using the wave variable ξ = k 1 x − ct reduce it to an ODE Integrating Eq. (36) with respect to ξ and considering the zero constants for integration, we obtain for simplicity suppose k 1 = 1.By a similar derivation as illustrated in the previous section, we obtain Therefore by use of the sech 2 method, we may choose a solution in the form Substituting Eq. (39) in to Eq. (37), and by using the well-known software Maple, and equating the coefficients of the powers Y , we then get the following algebraic relations: By using explicitly Eqs. ( 39) and (40) we solve this system with the aid of the Maple Package, we obtain the system of following results: Substituting Eq. (41) in Eq. (39) along with (11), we obtain exact travelling wave solution for Eq. ( 35) of the form: If we choose the following solution forms of Eqs. ( 15)-( 18) and insert them into Eq.(37), equating the coefficients of the powers Y , then we get the following algebraic relations: By the same manipulation as illustrated above, we obtain where c and µ are arbitrary constants.Substituting Eq. (44) in Eq. (39) along with (15), we obtain exact travelling wave solution for Eq.(35) of the form: which is the exact solution of symmetric regularized long wave (SRLW) equation.Can be seen that the results are the same, with comparing results Xu's [18].

Conclusion
In this article we investigated the symmetric regularized long wave (SRLW) equation by the analytical methods.Obtained the solitary wave and periodic wave solutions by the tanh-coth method and the sech 2 method.These methods have been successfully applied to obtain some new generalized solitonary solutions to the symmetric regularized long wave (SRLW) equation.The tanh-coth method and the sech 2 method are more powerful in searching for exact solutions of NLPDEs.