SOLITARY WAVE SOLUTIONS TO TIME-FRACTIONAL COUPLED DEGENERATE HAMILTONIAN EQUATIONS

The aim of this paper is to investigate the solution of timefractional coupled degenerate Hamiltonian equations. We use the G/G expansion method to determine soliton solutions of this system for different values of the fractional order. AMS Subject Classification: 26A33, 74J35, 35R11


Introduction and Discussion
Fractional order partial differential equations, as generalizations of classical integer order partial differential equations, are increasingly used to model problems in fluid, finance, physical and biological processes and systems [1,2,3,4].Consequently, considerable attention has been given to the solution of fractional ordinary differential equations, integral equations and fractional partial differential equations.Since most fractional differential equations do not have exact analytic solutions, approximation and numerical techniques, therefore, are used extensively.Recently, He and Lee [5,6,7] suggested a fractional complex trans-form instrument that converts fractional derivatives into classical derivatives.They considered the following PDE: where u denotes the modified Riemann-Liouville derivative [8,9,10]; 0 < α ≤ 1 and 0 < β ≤ 1.Then, the following transforms are introduced: where p and q are constants.
Using the above transforms, we can convert fractional derivatives into classical derivatives:

Preliminaries
Definitions for fractional derivatives: for more details see Ref. [8].In our paper we use the modified Riemann-Liouville derivative which was defined by Jumarie [9,10,11] which has merits over the original one, for example, the α-order derivative of a constant is zero.The main properties of the modified Riemann-Liouville derivative were summarized in [9] and three useful formulas of them are given as follows: In the literature, many significant methods have been proposed for obtaining exact solutions of nonlinear partial differential equations such as the tanh method, trigonometric and hyperbolic function methods, the rational sinecosine method, the extended tanh-function method, the Exp-function method, the Hirota's method , Hirota bilinear forms, the tanh-sech method , the first integral method and so on [12]- [28].In this work, we implement the G ′ /G expansion method [29,30,31,32,33] with the help of symbolic computation to derive soliton solution of the Hamiltonian system that reads

Survey of G ′ /G Method
Consider the following nonlinear partial differential equation: where u = u(x, t) is an unknown function, P is a polynomial in u = u(x, t) and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.By the wave variable ζ = x − ct the PDE (1) is then transformed to the ordinary differential equation (ODE) where u = u(ζ).Suppose that the solution of ODE (2) can be expressed by a polynomial in G ′ /G as follows [29,30,31,32,33] where G = G(ζ) satisfies the second order differential equation in the form a 0 , a 1 , ..., a m , λ and µ are constants to be determined later, provided that a m = 0.The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in the ODE (2).Now, if we let then by the help of ( 4) we get Or, equivalently By result (7) and implicit differentiation, one can derive the following two formulas Combining equations ( 3), ( 5) and (7-9), then it results in a polynomial of powers of Y .Then, collecting all terms of same order of Y and equating to zero, yields a set of algebraic equations for a 0 , a 1 , ..., a m , λ, and µ.
It is known that the solution of equation ( 4) is a linear combination of sinh and cosh or of sine and cosine, respectively, if △ = λ 2 − 4µ > 0 or △ < 0. Without lost of generality, we consider the first case and therefore

Time-Fractional Hamiltonian System
In this section, we study the time-fractional Hamiltonian system: The fractional transform s = t α Γ(1+α) reduces (1) to where u = u(s, x), v = v(s, x).The wave variable ζ = x − cs transforms the PDEs (2) into the following ODEs where From the first equation of (3) we have the relation and from the second equation of (3) and the relation ( 4) we reach to the following ODE 1 2 Integrating the obtained equation in ( 5) yields where R is the constant of integration.Assume the solution of ( 6) is Then, recalling (3) and (7) we have and Balancing the orders in ( 8) and ( 9), we require that m + 1 = 2m.Thus, m = 1, and therefore (7) can be rewritten as Now, we substitute equations ( 10) and ( 7) in (6) to get the following algebraic system: Solving the above system yields By the assumption given in (10) the function u(x, t) is given by provided that A 2 = B 2 .Now, by (4) the function v(x, t) is given by

Conclusion
In general, there exist no method that yields solitary solutions for fractional solitary wave equations.But, a fractional wave transform is adopted in this paper to convert such equations into classical partial differential equations.We succeeded in extracting solitary solutions for time-fractional Hamiltonian system.

Acknowledgement
The authors of this paper would like to thank the Editor and the anonymous referees for their in-depth reading, criticism of, and insightful comments on an earlier version of this paper.