Closed form solutions of complex wave equations via the modified simple equation method

Abstract The Kundu–Eckhaus equation and the derivative nonlinear Schrodinger equation describe various physical processes in nonlinear optics, plasma physics, fluid mechanics, magneto-hydrodynamic equation in the presence of the Hall Effect. Thus, closed form solutions of these equations are very important to realize the obscurity of the phenomena. The modified simple equation (MSE) method is highly effective and competent mathematical tool to examine closed form wave solutions of nonlinear evolution equations (NLEEs) arising in mathematical physics, applied mathematics and engineering. In this article, the MSE method is suggested and executed to construct closed form wave solutions of the above-mentioned equations involving parameters. When the parameters receive special values, impressive solitary wave solutions are derived from the exact solutions.

[1]  Norhashidah Hj. Mohd. Ali,et al.  Exp-Function Method for Duffing Equation and New Solutions of (2+1) Dimensional Dispersive Long Wave Equations , 2011 .

[2]  M. Akbar,et al.  Exact solutions to the Benney-Luke equation and the Phi-4 equations by using modified simple equation method , 2015 .

[3]  Farah Aini Abdullah,et al.  The exp-function method for new exact solutions of the nonlinear partial differential equations , 2011 .

[4]  J. Manafianheris.  Exact Solutions of the BBM and MBBM Equations by the Generalized (G'/G )-expansion Method Equations , 2012 .

[5]  林机,et al.  Exact Solutions of （2＋1）-Dimensional HNLS Equation , 2010 .

[6]  R. Hirota,et al.  Soliton solutions of a coupled Korteweg-de Vries equation , 1981 .

[7]  K. Khan,et al.  Application of Exp(- ( ))-expansion Method to Find the Exact Solutions of Modified Benjamin-Bona-Mahony Equation , 2013 .

[8]  Adem C. Cevikel,et al.  A procedure to construct exact solutions of nonlinear evolution equations , 2012 .

[9]  Anjan Biswas,et al.  Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation , 2008, Appl. Math. Comput..

[10]  Farah Aini Abdullah,et al.  New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method , 2012, J. Appl. Math..

[11]  Kwok Wing Chow,et al.  A class of exact, periodic solutions of nonlinear envelope equations , 1995 .

[12]  Zhenya Yan,et al.  New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water , 2001 .

[13]  R. Hirota.  Exact envelope‐soliton solutions of a nonlinear wave equation , 1973 .

[14]  Mingliang Wang.  SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS , 1995 .

[15]  Jianhong Wu.  Traveling Wave Solutions , 1996 .

[16]  Anjan Biswas,et al.  Modified simple equation method for nonlinear evolution equations , 2010, Appl. Math. Comput..

[17]  Exact travelling wave solutions for some nonlinear partial differential equations , 2010 .

[18]  Kongqing Yang,et al.  Exact solutions of nonlinear PDE, nonlinear transformations and reduction of nonlinear PDE to a quadrature , 2001 .

[19]  Md. Nur Alam,et al.  Study of Nonlinear Evolution Equations to Construct Traveling Wave Solutions via the New Approach of the Generalized (G'/G) -Expansion Method , 2013 .

[20]  Zuntao Fu,et al.  JACOBI ELLIPTIC FUNCTION EXPANSION METHOD AND PERIODIC WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS , 2001 .

[21]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[22]  Ji-Huan He.  AN ELEMENTARY INTRODUCTION TO RECENTLY DEVELOPED ASYMPTOTIC METHODS AND NANOMECHANICS IN TEXTILE ENGINEERING , 2008 .

[23]  Kamruzzaman Khan,et al.  Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method , 2013 .

[24]  Abdul-Majid Wazwaz,et al.  A sine-cosine method for handlingnonlinear wave equations , 2004, Math. Comput. Model..

[25]  Norhashidah Hj. Mohd. Ali,et al.  Abundant Exact Traveling Wave Solutions of Generalized Bretherton Equation via Improved (G′/G)-Expansion Method , 2012 .

[26]  A. Hendi.  THE EXTENDED TANH METHOD AND ITS APPLICATIONS FOR SOLVING NONLINEAR PHYSICAL MODELS , 2010 .

[27]  Lin Ji,et al.  GENERAL: Exact Solutions of (2+1)-Dimensional HNLS Equation , 2010 .

[28]  Jiao Zhang,et al.  An improved (G′/G)-expansion method for solving nonlinear evolution equations , 2010, Int. J. Comput. Math..

[29]  S. Champeaux,et al.  Remarks on the parallel propagation of small-amplitude dispersive Alfvénic waves , 1999 .

[30]  D. Levko,et al.  Modeling of Kundu-Eckhaus equation , 2007, nlin/0702050.

[31]  A. Bekir,et al.  Exact solutions for nonlinear evolution equations using Exp-function method , 2008 .

[32]  Shaolin Li,et al.  Exact Solutions of the Klein-Gordon Equation by Modified Exp-Function Method 1 , 2012 .

[33]  K. Khan,et al.  The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation: The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations , 2013 .

[34]  M. Ali Akbar,et al.  The alternative -expansion method with generalized Riccati equation: Application to fifth order (1+1)-dimensional Caudrey-Dodd-Gibbon equation , 2012 .

[35]  Sirendaoreji.  New exact travelling wave solutions for the Kawahara and modified Kawahara equations , 2004 .

[36]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[37]  Nassar H. Abdel-All,et al.  Expanding the Tanh-Function Method for Solving Nonlinear Equations , 2011 .

[38]  Nasir Taghizadeh,et al.  The first integral method to some complex nonlinear partial differential equations , 2011, J. Comput. Appl. Math..

[39]  Khaled A. Gepreel,et al.  On the solitary wave solutions for nonlinear Hirota–Satsuma coupled KdV of equations , 2004 .

[40]  E. Zayed,et al.  Applications of an Extended (′/)-Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics , 2010 .

[41]  S. Ibrahim,et al.  Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics Using the Modified Simple Equation Method , 2012 .

[42]  E. Fan,et al.  Extended tanh-function method and its applications to nonlinear equations , 2000 .

[43]  K. Porsezian,et al.  Soliton perturbation theory for the modified nonlinear Schrödinger’s equation , 2007 .

[44]  Qi Wang,et al.  Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation , 2005 .

[45]  Mingliang Wang,et al.  The periodic wave solutions for the Klein–Gordon–Schrödinger equations , 2003 .