New Explicit Exact Solutions of Nonlinear Evolution Equations Using the Generalized Auxiliary Equation Method Combined with Exp-function Method

New Explicit Exact Solutions of Nonlinear Evolution Equations Using the Generalized Auxiliary Equation Method Combined with Exp-function Method ~ The investigation of exact solutions of nonlinear evolution equations plays an important role in the study of
nonlinear physical phenomena and gradually becomes one of the most important and significant tasks. In the past several decades, many effective methods for obtaining exact solutions of NLEEs have been presented [1 − 30].
In recent years, the direct search for exact solutions of PDEs has become more and more attractive partly due to the availability of computer symbolic systems like Maple or Mathematica, which allows us to perform the complicated and tedious algebraic calculations on computer.
Very recently, He and Abdou [22], Abdou [23 − 25] proposed a straightforward and concise method, called Exp-function method, to obtain generalized solitary solutions and periodic solutions of NLEEs. The solution procedure of this method, by the help of Maple, is of utter simplicity and this method can be easily extended to other nonlinear evolution equations. The Exp-function is more general than the sinh-function and the tanh-function, so we can found more general solutions in the Exp-function method. The solution procedure, using Matab or Mathematica, is of utter simplicity. The Exp-function method can be employed in both the straightforward way and the sub-equation way. But we suggest that it is better to use this method directly, not only for its convenience, but also because it is sometimes possible to lose some information and solutions if we apply it in the subequation way. The Exp-function method is more convenient and effective than the extended Fan sub-equation method. In the present paper is to extend the Exp-function method [22 − 26] to generalized auxiliary equation. We consider the generalized auxiliary equation

φ 2 (ξ) = E + Pφ4(ξ) + Rφ3(ξ) + Qφ2(ξ)

where P, Q, R and E are constants to be determined later. Then employ the generalized auxiliary Eq.(1) and its generalized solitary solutions to find new and more general exact solutions of two nonliner evolution equations, namely, the higher-order nonlinear Schrodinger equation and generalized Zakharov equations.

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