Generalized and Improved (G9/G)-Expansion Method for (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation

Transcription

1 for (3+)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation Hasibun Naher * Farah Aini Abdullah M. Ali Akbar 3 School of Mathematical Sciences Universiti Sains Malaysia Penang Malaysia Department of Mathematics and Natural Sciences BRA University Dhaka Bangladesh 3 Department of Applied Mathematics University of Rajshahi Rajshahi Bangladesh Abstract The generalized and improved ðg 0 =GÞ-expansion method is a powerful and advantageous mathematical tool for establishing abundant new traveling wave solutions of nonlinear partial differential equations. In this article we investigate the higher dimensional nonlinear evolution equation namely the (3+)-dimensional modified KdV-Zakharov-Kuznetsev equation via this powerful method. The solutions are found in hyperbolic trigonometric and rational function form involving more parameters and some of our constructed solutions are identical with results obtained by other authors if certain parameters take special values and some are new. The numerical results described in the figures were obtained with the aid of commercial software Maple. itation: Naher H Abdullah FA Akbar MA (03) Generalized and Improved (G9/G)-Expansion Method for (3+)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation. PLoS ONE 8(5): e6468. doi:0.37/journal.pone Editor: Francesco Pappalardo University of atania Italy Received November 6 0; Accepted April 6 03; Published May 3 03 opyright: ß 03 Naher et al. This is an open-access article distributed under the terms of the reative ommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original author and source are credited. Funding: This work was supported by Universiti Sains Malaysia (USM) short term grant (Ref. No. 304/PMATHS/63007). The funder had no role in study design data collection and analysis decision to publish or preparation of the manuscript. ompeting Interests: The authors have declared that no competing interests exist. * hasibun06tasauf@gmail.com Introduction Nonlinear partial differential equations (PDEs) are widely used to describe complex physical phenomena in different branches of mathematical physics engineering sciences and other technical arenas. The analytical solutions of nonlinear evolution equations (NLEEs) have now become a more exciting topic for a diverse group of scientists. In recent years they established several powerful methods to obtain exact solutions. For example the Backlund transformation method [] the inverse scattering method [3] the truncated Painleve expansion method [4] the Weierstrass elliptic function method [5] the Hirota s bilinear transformation method [6] the Jacobi elliptic function expansion method [7 9] the generalized Riccati equation method [0] the tanh-coth method [ 3] the F-expansion method [45] the direct algebraic method [6] the Exp-function method [7 3] and others [4 8]. Every method has some restrictions in their implementations. Basically there is no integrated method which could be utilized to handle all types of nonlinear PDEs. Another powerful and effective method has been presented by Wang et al. [9] to construct exact traveling wave solutions and called the ðg =GÞ-expansion method. In this method they employed the second order linear ordinary differential equation (ODE) G zlg zmg~0 where l and m are arbitrary constants. Afterwards several researchers applied this basic method to obtain traveling wave solutions for different nonlinear PDEs [30 33]. Recently Zhang et al. [34] extended the ðg =GÞ-expansion method which is called the improved ðg =GÞ- expansion method. In this method u(j)~ Xm a i (G =G) i is used i~{m as traveling wave solutions where either a {m or a m may be zero but both a {m and a m cannot together be zero. And a diverse group of scientists implemented this method to establish new traveling wave solutions of NLEEs [35 38]. Very recently Akbar et al. [39] extended and improved this method by using bðjþ~ Pm e {n n as traveling wave ðdzðg n~ {m 0 =GÞÞ solutions where either e {m or e m may be zero but both e {m and e m cannot be zero together and is called the generalized and improved ðg =GÞ-expansion method. This method could be applied for generating a rich class of traveling wave solutions because in this method an additional variable d is applied and it can produce more general and abundant solutions. If d~0 we can obtain the same solutions according to Zhang et al. [34]. The aim of this work is that we concentrate to find more general and abundant traveling wave solutions of the (3+)- dimensional modified KdV-Zakharov-Kuznetsev equation by implementing the generalized and improved ðg 0 =GÞ-expansion method. Description of the Method Let us consider a general nonlinear PDE: Auu t u x u y u z u xt u yt u zt u xy u tt u xx u yy ::: ~0 ðþ where u~uðxyztþ is an unknown function A is a polynomial in its arguments and the subscripts stand for the partial derivatives. The main steps of the method [39] are as follows: PLOS ONE May 03 Volume 8 Issue 5 e6468

2 Step. We suppose the traveling wave variable: uxyzt ð Þ~bðjÞ j~ xzyzz+v t ðþ u t zbu u x z u xxx zu xyy zu xzz ~0 ð7þ where V is the speed of the traveling wave. Using Eq. () Eq. () is converted into an ordinary differential equation for bðjþ : Now we use the wave transformation Eq. () into the Eq. (7) which yields: Bbb ð 0 b b ::: Þ~0 ð3þ {Vb zbb b z3b ~0 ð8þ where the superscripts indicate the ordinary derivatives with respect to j: Step. According to possibility Eq. (3) can be integrated term by term one or more times yielding constant(s) of integration. The integral constant may be zero for simplicity. Step 3. Suppose that the traveling wave solution of Eq. (3) can be expressed by a polynomial in ðdzðg 0 =GÞÞ as follows: bðjþ~ Xm n~ {m e {n ðdzðg 0 =GÞÞ n ð4þ where either e {m or e m may be zero but both e {m and e m cannot be zero simultaneously e n ðn~0+ + :::+ mþ and d are arbitrary constants to be determined later and G~GðjÞsatisfies the following second order linear ODE: G zlg zmg~0 where l and m are constants. Step 4. To determine the positive integer m taking the homogeneous balance between the highest order nonlinear terms and the highest order derivatives appearing in Eq. (3). If the degree of bðjþ is Dbj ½ ð ÞŠ~m then the degree of the other expression would be as follows: D dp bðjþ dj p ~mzp D b p d q bðjþ dj q s ð5þ ~mpzsðmzqþ ð6þ Step 5. Substituting Eq. (4) and Eq. (5) into Eq. (3) together with the value of m obtained in Step 4 yields polynomials in ðdzg 0 =GÞ m and ðdzg 0 =GÞ {m ðm~0 ::: Þ: ollecting each coefficient of the resulted polynomials to zero we obtain a set of algebraic equations for e n ðn~0+ + :::+ mþ d and V: Step 6. Suppose that the value of the constants e n ðn~0+ + :::+mþ d and Vcan be found by solving the algebraic equations which are obtained in step 5. Since the general solution of Eq. (5) is well known to us substituting the values of e n ðn~0+ + :::+mþ d and V into Eq. (4) we can obtain more general type and new exact traveling wave solutions of the nonlinear partial differential equation (). Application of the Method In this section we apply the generalized and improved ðg 0 =GÞ- expansion method to establish more general and some new exact traveling wave solutions of the well known (3+)-dimensional modified KdV-Zakharov-Kuznetsev equation. Let us consider the (3+)-dimensional modified KdV-Zakharov- Kuznetsev equation followed by Zayed [40]: Eq. (8) is integrable therefore integrating with respect j once yields: {Vbz 3 bb3 z3b ~0 where is an integral constant which is to be determined later. Taking the homogeneous balance between b 3 and b in Eq. (9) we obtain m~. Therefore the solution of Eq. (9) is of the form: ð9þ bðjþ~e { ðdzg 0 =GÞ { ze 0 ze ðdzg 0 =GÞ ð0þ where e { e 0 and e are constants to be determined. Substituting Eq. (0) together with Eq. (5) into Eq. (9) the lefthand side is converted into polynomials in ðdzg 0 =GÞ m and ðdzg 0 =GÞ {m ðm~0 ::: Þ: We collect each coefficient of these resulting polynomials to zero yielding a set of simultaneous algebraic equations (for simplicity which are not presented) for e { e 0 e d and V: Solving these algebraic equations with the help of symbolic computation system Maple 3 we obtain following. ase : e { ~0 e 0 ~e 0 e ~+ p 6i ffiffiffiffiffi ~0 V~ {3 l {4m d~ l+ ie pffiffiffiffiffi 0 3 where e 0 l and m are free parameters. ase : e { ~+ 6id 3i p ffiffiffiffiffi e 0 ~+ ð {lzd p ffiffiffiffiffi Þ e ~0 ~0 V~ {3 l {4m d~d where e 0 d l and m are free parameters. ase 3: ðþ ðþ e { ~+ 3i l {4m p ffiffiffiffiffi e 0 ~0 e ~+ p 6i ffiffiffiffiffi ~0 V~ 3 ð3þ l {4m ð{+3þ d~ l where e 0 l and m are free parameters. PLOS ONE May 03 Volume 8 Issue 5 e6468

3 ase 4: V~8dðl{dÞ{ 3 l z8m 3i {lzd e 0 ~+ ð p ffiffiffiffiffi Þ e ~+ p 6i ffiffiffiffiffi d~d 36i {lzd ~+ ð Þ d p ffiffiffiffiffi e{ ~+ 6id p ffiffiffiffiffi ð4þ where e 0 d l and m are free parameters. Substituting the general solution Eq. (5) into Eq. (0) we obtain the following. When l {4mw0 we obtain following hyperbolic function solution: When l {4m~0 we obtain the rational function solution: bðjþ~e { dz {l z D { ze 0 z zdj e dz {l z D zdj ð7þ For case substituting Eq. () into Eq. (5) and simplifying yields following traveling wave solutions when ~0 but D=0 and D~0 but =0) respectively: s l {4m b ðjþ~e 0 +e 0 +3i coth l {4m j bðjþ~e l {ðd{ z sinh l l {4m jzd cosh l {4m j {4m A cosh l {4m jzd sinh l {4m j { ð5þ s l {4m b ðjþ~e 0 +e 0 +3i tanh l {4m j Again substituting Eq. () into Eq. (6) and simplifying our exact solutions become when ~0 but D=0 and D~0 but =0 respectively: ze 0 ze ðd{ l z sinh l l {4m jzd cosh l {4m j {4m A cosh l {4m jzd sinh l {4m j where and D are arbitrary constants if and D take particular values various known solutions can be rediscovered. When l {4mv0 we obtain the trigonometric function solution: bðjþ~e l {ðd{ z { sin 4m{l 4m{l jzd cos 4m{l j A cos 4m{l jzd sin 4m{l j ze 0 ze ðd{ l z { sin 4m{l 4m{l jzd cos 4m{l j A cos 4m{l jzd sin 4m{l j { ð6þ s 4m{l b 3 ðjþ~e 0 +e 0 +3i cot 4m{l j s 4m{l b 4 ðjþ~e 0 +e 0 +3i tan 4m{l j Moreover substituting Eq. () into Eq. (7) and simplifying our obtained solutions becomes: b 5 ðjþ~e 0 +e 0 + 6i D zdj where j~ x zyz zz 3 l {4m t: Similarly for case substituting Eq. () into Eq. (5) and simplifying yields following traveling wave solutions when ~0 but D=0; D~0 but =0 respectively: b ðjþ~+ 3i pffiffiffiffiffi d 0 d{ l z l {4m l {4m ja { zðl{d ÞA where and D are arbitrary constants if and D take particular values various known solutions can be rediscovered. PLOS ONE 3 May 03 Volume 8 Issue 5 e6468

4 Table. omparison between Naher et al. [4] solutions and Newly obtained solutions. Naher et al. [4] New solutions i.if l~4m~3 and b~ solution Eq. (5) becomes: vðgþ~+3i coth g: ii. If l~6m~7 and b~4 solution Eq. (6) becomes: pffiffiffi vðgþ~+3i tanh g i. If e 0 ~l~4m~3b~ and b ðjþ~vðgþ solution b ðjþbecomes: vðgþ~+3i coth g: ii. If e 0 ~l~6m~7b~4 and b ðjþ~vðgþ solution b ðjþbecomes: pffiffiffi : vðgþ~+3i tanh g : iii. If l~4m~5 and b~8 solution Eq. (9) becomes: vðgþ~ +3i cot g: iii. If e 0~3l~4m~5b~8 and b 3 ðjþ~vðgþ solution b 3 ðjþ becomes: vðgþ~ +3i cot g: iv. If l~6m~0 and b~8 solution Eq. (0) becomes: vðgþ~+ i tan g: v. If l~m~a~b~ and b~8 solution Eq. () becomes: i vðgþ~+ ð Þ : i vðgþ~+ zðxzyzzþ v. If e 0 ~l~m~~d~ and b 5 j z xzyzz doi:0.37/journal.pone t00 iv. If e 0 ~l~6m~0b~8 and b 4 ðjþ~vðgþ solution b 4 ðjþ becomes: vðgþ~+ i tan g: ð Þ~vðgÞ solution b 5 ðjþ becomes: b ðjþ~+ p 0 d{ l z ð d l {4m tanh l {4m ja { zðl{d ÞA Substituting Eq. () into Eq. (6) and simplifying yields following traveling wave solutions when ~0 but D=0; D~0 but =0 respectively: b 3 ðjþ~+ p 0 d{ l z ð d 4m{l cot 4m{l ja { zðl{d ÞA s b3 ðjþ~+6i csc h l l {4m {4m j Substituting Eq. (3) into Eq. (6) and simplifying yields following traveling wave solutions when ~0 but D=0; D~0 but =0 respectively: s b3 ðjþ~+6i csc 4m{l 4m{l j Substituting Eq. (3) into Eq. (7) and simplifying yields following traveling wave solutions: b3 3 ðjþ~+ p 6i ffiffiffiffiffi! D zdj { l {4m D { 4 zdj where j~ x zyz zz 3 l {4m ð+3þ t: b 4 ðjþ~+ p 0 d{ l { ð d 4m{l tan 4m{l ja { zðl{d ÞA Finally substituting Eq. () into Eq. (7) and simplifying yields following traveling wave solutions: b 5 ðjþ~+ p! d l D { d{ zðl{dþ zdj where j~ x zyz zz 3 l {4m t: Again for case 3 substituting Eq. (3) into Eq. (5) and simplifying yields following traveling wave solutions when ~0 but D=0; D~0 but =0 respectively: Figure. Solitons solution for b 4. l~4 m~4:5 b~9 e 0 ~0:5. doi:0.37/journal.pone g00 PLOS ONE 4 May 03 Volume 8 Issue 5 e6468

5 Figure. Solitons solution for b 4 4. l~7 m~:5 b~9 d~3:5. doi:0.37/journal.pone g00 Moreover for case 4 substituting Eq. (4) into Eq. (5) and simplifying yields following traveling wave solutions (if ~0 but D=0; D~0 but =0) respectively: 0 b4 ðjþ~+ p 6i ffiffiffiffiffið d l d{ z l coth l {4m jþ { l {4m z coth l {4m jþ Figure 4. Solitons solution for b3. l~4 m~4:5 b~ e 0 ~0:5. doi:0.37/journal.pone g004 0 b4 ðjþ~+ 6i pffiffiffiffiffið l d{ z l {4m l {4m l {4m l {4m jþ tanh jþ { z tanh for case 4 substituting Eq. (4) into Eq. (6) and simplifying yields following traveling wave solutions when ~0 but D=0; D~0 but =0 respectively: Figure 3. Periodic solution forb. l~3 m~ b~ e 0 ~. doi:0.37/journal.pone g003 Figure 5. Periodic solution for b3 3. l~4 m~4 b~5 ~5 D~9. doi:0.37/journal.pone g005 PLOS ONE 5 May 03 Volume 8 Issue 5 e6468

6 Figure 6. Solitons solution for b 4. l~7 m~:5 b~3 e 0 ~. doi:0.37/journal.pone g006 0 b4 3 ðjþ~+ 6i pffiffiffiffiffið l d{ z cot 4m{l jþ { 4m{l z cot q 0 b4 4 ðjþ~+ 6i pffiffiffiffiffið l d{ { 4m{l q 4m{l jþ 4m{l tan 4m{l jþ { 4m{l z tan 4m{l jþ for case 4 substituting Eq. (4) into Eq. (7) and simplifying yields following traveling wave solutions: Figure 8. Solitons solution for b 4. l~4 m~4:5 b~9 d~. doi:0.37/journal.pone g008 b4 5 ðjþ~+ 6i pffiffiffiffiffi d l D { d{ zdj z D zdj where j~ x zyz zz 3 l z8m {8dðl{dÞ t: Results and Discussion The higher dimensional modified KdV-Zakharov-Kuznetsev equation has been solved by many authors by implementing different methods. For example Zayed [40] executed the basic ðg 0 =GÞ-expansion method Xu [4] used the elliptic equation method Naher et al. [8] applied the Exp-function method furthermore they [4] employed the improved ðg 0 =GÞ-expansion method to obtain traveling wave solutions of this mentioned equation. But in this article we construct more general and new exact traveling wave solutions by applying the generalized and improved ðg 0 =GÞ-expansion method with an additional free parameter d: The obtained solutions would be useful to understand the mechanism of the complicated nonlinear physical phenomena in a wave interaction. Moreover some solutions are identical with already published results which are described in table. Beyond this table we obtain new exact solutions b b b 3 b 4 b 5 b3 b3 b3 3 b4 b4 b4 3 b4 4 and b4 5 which are not established in the previous literature. Also solutions b b 4 b3 b3 3 b4 and b4 4 are depicted in Figures Graphical Presentations of Some Solutions The graphical presentations of some solutions are illustrated in Figures with the aid of commercial software Maple. Figure 7. Solitons solution for b 4 4. l~6 m~9:5 b~9 d~3. doi:0.37/journal.pone g007 onclusions In this article the generalized and improved ðg 0 =GÞ-expansion method is implemented to produce plentiful new traveling wave solutions of the (3+)-dimensional modified KdV-Zakharov- Kuznetsev equation. The used method has many advantages: it PLOS ONE 6 May 03 Volume 8 Issue 5 e6468

7 is straightforward and concise. Further the obtained solutions reveal that this method is a promising mathematical tool because it can furnish a different class of new traveling wave solutions with free parameter of distinct physical structures. Subsequently this prominent method could be more effectively used to solve various nonlinear partial differential equations which regularly arise in science engineering and other technical arenas. Acknowledgments The authors would like to express their earnest thanks to the anonymous referee(s) for their useful and valuable comments and suggestions. References. Mimura MR (978) Backlund Transformation. Springer Germany.. Rogers Shadwick WF (98) Backlund Transformations and their applications. Academic Press New York. 3. Ablowitz MJ larkson PA (99) Solitons nonlinear evolution equations and inverse scattering transform. ambridge Univ. Press ambridge. 4. Weiss J Tabor M arnevale G (98) The painleve property for partial differential equations. Journal of Mathematical Physics 4: Kudryashov NA (990) Exact solutions of the generalized Kuramoto-Sivashinsky equation. Physics Letters A 47: Hirota R (97) Exact solution of the Korteweg-de-Vries equation for multiple collisions of solutions. Physical Review Letters 7: Liu S Fu Z Liu S Zhao Q (00) Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Physics Letters A 89: hen Y Wang Q (005) Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (+)- dimensional dispersive long wave equation. haos Solitons and Fractals 4: Lu D (005) Jacobi elliptic function solutions for two variant Boussinesq equations. haos Solitons and Fractals 4: Yan Z Zhang HQ (00) New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water. Physics Letters A 85: Malfliet W (99) Solitary wave solutions of nonlinear wave equations. American Journal of Physics 60: Wazwaz AM (007) The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. Applied Mathematics and omputation 88: Bekir A evikel A (009) Solitary wave solutions of two nonlinear physical models by tanh-coth method. ommun Nonlinear Sci Numer Simulat 4: Wang ML Li XZ (005) Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. haos Solitons and Fractals 4: Abdou MA (007) The extended F-expansion method and its application for a class of nonlinear evolution equations. haos Solitons and Fractals 3: Soliman AA Abdou HA (009) New exact Solutions of nonlinear variants of the RLW the PHI-four and Boussinesq equations based on modified extended direct algebraic method. International Journal of Nonlinear Science 7: He JH Wu XH (006) Exp-function method for nonlinear wave equations. haos Solitons and Fractals 30: Naher H Abdullah FA Akbar MA (0) New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method. Journal of Applied Mathematics Article ID: pages doi: 0.55/0/ Mohyud-Din ST Noor MA Noor KI (00) Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation. Journal of King Saud University. : Naher H Abdullah FA Akbar MA (0) The exp-function method for new exact solutions of the nonlinear partial differential equations. International Journal of the Physical Sciences 6: Yildirim A Pinar Z (00) Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology. omputers & Mathematics with Applications 60: Aslan I (0) Application of the exp-function method to nonlinear lattice differential equations for multi-wave and rational solutions. Mathematical Methods in the Applied Sciences Doi: 0.00/mma Ma WX Zhu Z (0) Solving the (3+)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm Applied Mathematics and omputation 8: Author ontributions onceived and designed the experiments: HN FAA MAA. Performed the experiments: HN FAA MAA. Analyzed the data: HN FAA MAA. ontributed reagents/materials/analysis tools: HN FAA MAA. Wrote the manuscript: HN. Designed the software used in analysis generating solutions visualizing some of the solutions: HN. Discussed and polished the manuscript: FAA MAA. Read and approved the manuscript: HN FAA MAA. 4. Triki H Biswas A (0) Soliton solutions for a generalized fifth-order KdV equation with t-dependent coefficients. Waves in Random and omplex Media : Wazwaz AM (0) A new (+)-dimensional Korteweg-de-Vries equation and its extension to a new (3+)-dimensional Kadomtsev-Petviashvili equation. Physica Scripta doi:0.088/ /84/03/ Noor M Noor K Waheed A Al-Said EA (0) An efficient method for solving system of third-order nonlinear boundary value problems. Mathematical Problems in Engineering Article ID pages doi: 0.55/0/ Weise LD Nash MP Panfilov AV (0) A discrete model to study reactiondiffusion mechanics systems. Plos One 6(7): e934. doi: 0.37/journal.- pone Yang J Liang S Zhang Y (0) Travelling waves of delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion. Plos One 6(6): e8. doi: 0.37/journal.pone Wang M Li X Zhang J (008) The (G9/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A 37: Feng J Li W Wan Q (0) Using (G9/G)-expansion method to seek traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation. Applied Mathematics and omputation 7: Naher H Abdullah FA Akbar MA (0) The (G9/G)-expansion method for abundant traveling wave solutions of audrey-dodd-gibbon equation. Mathematical Problems in Engineering Article ID: 86 pages doi:0.55/0/ Abazari R Abazari R (0) Hyperbolic trigonometric and rational function solutions of Hirota-Ramani equation via (G9/G)-expansion method. Mathematical Problems in Engineering Article ID: 4480 pages doi:0.55/0/ Zayed EME Al-Joudi S (00) Applications of an Extended (G9/G)-Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics. Mathematical Problems in Engineering Article ID pages doi:0.55/00/ Zhang J Jiang F Zhao X (00) An improved (G9/G)-expansion method for solving nonlinear evolution equations. International Journal of omputer Mathematics 87: Zhao YM Yang YJ Li W (0) Application of the improved (G9/G)-expansion method for the Variant Boussinesq equations. Applied Mathematical Sciences 5: Nofel TA Sayed M Hamad YS Elagan SK (0) The improved (G9/G)- expansion method for solving the fifth-order KdV equation. Annals of Fuzzy Mathematics and Informatics 3: Hamad YS Sayed M Elagan SK El-Zahar ER (0) The improved (G9/G)- expansion method for solving (3+)-dimensional potential-ytsf equation. Journal of Modern Methods in Numerical Mathematics : Naher H Abdullah FA (0) Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G9/G)-expansion method. Mathematical Problems in Engineering Article ID: pages doi:0.55/0/ Akbar MA Ali NHM Zayed EME (0) A generalized and improved (G9/G)- expansion method for nonlinear evolution equations. Mathematical Problems in Engineering Article ID: pages doi:0.55/0/ Zayed EME (00) Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G9/G)-expansion method. Journal of Applied Mathematics & Informatics 8: Xu G (006) An elliptic equation method and its applications in nonlinear evolution equations. haos Solitons and Fractals 9: Naher H Abdullah FA Akbar MA (0) New traveling wave solutions of the higher dimensional nonlinear evolution equation by the improved (G9/G)- expansion method. World Applied Sciences Journal 6:. PLOS ONE 7 May 03 Volume 8 Issue 5 e6468