İsmail Aslan

Analytic study on two nonlinear evolution equations by using the (G′/G)-expansion method

Abstract The validity and reliability of the so-called (G′/G)-expansion method is tested by applying it to two nonlinear evolutionary equations. Solutions in more general forms are obtained. When the parameters are taken as special values, it is observed that the previously known solutions can be recovered. New rational function solutions are also presented. Being concise and less restrictive, the method can be applied to many nonlinear partial differential equations.

Application of the G′/G-expansion method to Kawahara type equations using symbolic computation

Abstract In this paper, Kawahara type equations are selected to illustrate the effectiveness and simplicity of the G ′ / G -expansion method. With the aid of a symbolic computation system, three types of more general traveling wave solutions (including hyperbolic functions, trigonometric functions and rational functions) with free parameters are constructed. Solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters, involved in the traveling waves, as special values.

A note on the (G′/G)-expansion method again

Abstract We report an observation on two recent analytic methods; the ( G ′/ G )-expansion method and the simplest equation method.

Some Remarks on Exp-Function Method and Its Applications

Recently, many important nonlinear partial differential equations arising in the applied physical and mathematical sciences have been tackled by a popular approach, the so-called Exp-function method. In this paper, we present some shortcomings of this method by analyzing the results of recently published papers. We also discuss the possible improvement of the effectiveness of the method.

On the application of the Exp-function method to the KP equation for N-soliton solutions

Abstract We observe that the form of the Kadomstev–Petviashvili equation studied by Yu (2011) [S. Yu, N -soliton solutions of the KP equation by Exp-function method, Appl. Math. Comput. (2011) doi:10.1016/j.amc.2010.12.095] is incorrect. We claim that the N -soliton solutions obtained by means of the basic Exp-function method and some of its known generalizations do not satisfy the equation considered. We emphasize that Yu’s results (except only one) cannot be solutions of the correct form of the Kadomstev–Petviashvili equation. In addition, we provide some correct results using the same approach.

Exact and explicit solutions to nonlinear evolution equations using the division theorem

Abstract In this paper, we show the applicability of the first integral method, which is based on the ring theory of commutative algebra, to the regularized long-wave Burgers equation and the Gilson–Pickering equation under a parameter condition. Our method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are derived in a concise manner.

The Exp-function approach to the Schwarzian Korteweg-de Vries equation

By means of the Exp-function method and its generalization, we report further exact traveling wave solutions, in a concise form, to the Schwarzian Korteweg-de Vries equation which admits physical significance in applications. Not only solitary and periodic waves but also rational solutions are observed.

The Ablowitz–Ladik lattice system by means of the extended (G′/G)-expansion method

Abstract We analyzed the Ablowitz–Ladik lattice system by using the extended ( G ′ / G )-expansion method. Further discrete soliton and periodic wave solutions with more arbitrary parameters are obtained. We observed that some previously known results can be recovered by assigning special values to the arbitrary parameters.

Some Remarks on Exp-Function Method and Its Applications-A Supplement

Recently, the authors of [Commun. Theor. Phys. 56 (2011) 397] made a number of useful observations on Exp-function method. In this study, we focus on another vital issue, namely, the misleading results of double Exp-function method.

Exact solutions of a fractional-type differential-difference equation related to discrete MKdV equation

The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 (2011) 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters: hyperbolic, trigonometric and rational, which have not been reported before.