Many Exact Solutions for a Higher-Order Nonlinear Schrödinger Equation with Non-Kerr Terms Describing the Propagation of Femtosecond Optical Pulses in Nonlinear Optical Fibers

In this article, we apply two powerful methods, namely the first integral method and a direct algebraic method for constructing many exact solutions for the higher-order nonlinear Schrödinger equation with non-Kerr terms that describes the propagation of femtosecond optical pulses in nonlinear optical fibers. Using a simple transformation, we reduce the given equation to a nonlinear ordinary differential equation (ODE). Various solutions of the resulting nonlinear ODE are obtained by using the above two methods. A comparison between our recent results and the well-known results is given.

[1]  Emmanuel Yomba,et al.  A generalized auxiliary equation method and its application to nonlinear Klein–Gordon and generalized nonlinear Camassa–Holm equations , 2008 .

[2]  Sirendaoreji.  Auxiliary equation method and new solutions of Klein-Gordon equations , 2007 .

[3]  Ming-Liang Wang,et al.  The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations , 2010 .

[4]  Wen-Xiu Ma,et al.  Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm , 2012, Appl. Math. Comput..

[5]  Gui-qiong Xu.  Extended Auxiliary Equation Method and Its Applications to Three Generalized NLS Equations , 2014 .

[6]  E. Zayed,et al.  A further improved (G′/G)- expansion method and the extended tanh- method for finding exact solutions of nonlinear PDEs , 2011 .

[7]  Shi Jun-jie,et al.  Propagating Optical Phonon Modes and Their Electron-Phonon Interaction Hamiltonians in Asymmetric Wurtzite Nitride Semiconductor Quantum Wells , 2006 .

[8]  Xiangzheng Li,et al.  Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms , 2007 .

[9]  E. Zayed,et al.  The -Expansion Method and Its Applications for Solving Two Higher Order Nonlinear Evolution Equations , 2014 .

[10]  S. Ibrahim,et al.  The two variable (G'/G,1/G) -expansion method for finding exact traveling wave solutions of the (3+1) - dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation , 2013 .

[11]  Zhaosheng Feng,et al.  The first-integral method to study the Burgers–Korteweg–de Vries equation , 2002 .

[12]  A. H. Arnous,et al.  DNA Dynamics Studied Using the Homogeneous Balance Method , 2012 .

[13]  Xiangzheng Li,et al.  Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation , 2007 .

[14]  E. Zayed,et al.  Exact Solutions for The Generalized Zakharov-Kuznetsov Equation with Variable Coefficients Using The Generalized ((G'/G))-expansion Method , 2010 .

[15]  Elsayed M. E. Zayed,et al.  Some applications of the (G'/G)-expansion method to non-linear partial differential equations , 2009, Appl. Math. Comput..

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

[17]  Wenxiu Ma,et al.  A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation , 2009, 0903.5337.

[18]  Hongqing Zhang,et al.  Travelling wave solutions of nonlinear partial equations by using the first integral method , 2010, Appl. Math. Comput..

[19]  Xiangzheng Li,et al.  Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations , 2005 .

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

[21]  Wei Wang,et al.  A generalized (G′G)-expansion method for the mKdV equation with variable coefficients , 2008 .

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

[23]  Khaled A. Gepreel,et al.  The (G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics , 2009 .

[24]  B. Zheng.  Traveling wave solutions for some nonlinear evolution equations by the first integral method , 2011 .

[25]  S D Zhu,et al.  Exp-function Method for the Discrete mKdV Lattice , 2007 .

[26]  A. Bekir,et al.  Analytic treatment of nonlinear evolution equations using first integral method , 2012 .

[27]  B. Duffy,et al.  An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations , 1996 .

[28]  Y. A. Amer,et al.  The first integral method and its application for finding the exact solutions of nonlinear fractional partial differential equations (PDES) in the mathematical physics , 2014 .

[29]  Elsayed M. E. Zayed,et al.  Traveling Wave Solutions of the Nonlinear (3 + 1)-Dimensional Kadomtsev-Petviashvili Equation Using the Two Variables (G'/G, 1/G)-Expansion Method , 2012, J. Appl. Math..

[30]  E. Zayed,et al.  The ( $\frac{G'}{G})$ -expansion method and its applications to some nonlinear evolution equations in the mathematical physics , 2009 .

[31]  M. A. M. Abdelaziz,et al.  The Two-Variable (′/,1/)-Expansion Method for Solving the Nonlinear KdV-mKdV Equation , 2012 .

[32]  Xiangzheng Li,et al.  A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms , 2007 .

[33]  Ji-Huan He,et al.  Exp-function method for nonlinear wave equations , 2006 .

[34]  Mingliang Wang,et al.  The (G' G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics , 2008 .

[35]  E. Zayed,et al.  New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized -expansion method , 2009 .

[36]  Liu Shi-da,et al.  A New Approach to Solve Nonlinear Wave Equations , 2003 .

[37]  Wenxiu Ma,et al.  Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation , 1995, solv-int/9511005.

[38]  Emmanuel Yomba,et al.  The modified extended Fan sub-equation method and its application to the (2+1)-dimensional Broer–Kaup–Kupershmidt equation , 2006 .

[39]  Zhenya Yan,et al.  Abundant families of Jacobi elliptic function solutions of the (2+1)-dimensional integrable Davey–Stewartson-type equation via a new method , 2003 .

[40]  Mingliang Wang.  Exact solutions for a compound KdV-Burgers equation , 1996 .

[41]  N. Taghizadeh,et al.  Exact solutions of the nonlinear Schrödinger equation by the first integral method , 2011 .

[42]  M Lakshmanan,et al.  Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[43]  Mingliang Wang,et al.  The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations , 2004 .

[44]  Mingliang Wang,et al.  Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation , 2005 .

[45]  Wang Ming-liang,et al.  Extended F-Expansion Method and Periodic Wave Solutions for Klein–Gordon–Schrödinger Equations , 2006 .

[46]  Sirendaoreji.  A new auxiliary equation and exact travelling wave solutions of nonlinear equations , 2006 .

[47]  E. Zayed,et al.  The ()-Expansion Method and Its Applications to Find the Exact Solutions of Nonlinear PDEs for Nanobiosciences , 2014 .

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

[49]  Zhenya Yan.  New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations , 2001 .

[50]  T. Xia,et al.  A generalized new auxiliary equation method and its applications to nonlinear partial differential equations , 2007 .

[51]  Mingliang Wang,et al.  Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics , 1996 .

[52]  Xiangzheng Li,et al.  The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation , 2006 .

[53]  Elsayed M. E. Zayed,et al.  Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine-cosine and the exp-function methods , 2011, Appl. Math. Comput..

[54]  Sirendaoreji.  Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations , 2007 .