On solving two higher-order nonlinear PDEs describing the propagation of optical pulses in optic fibers using the \(\left( \frac{G^{\prime }}{G},\frac{1}{G}\right) \) -expansion method

详细信息    查看全文

• 作者：E. M. E. Zayed ; K. A. E. Alurrfi
• 关键词：The two variable $$({\frac{G^{\prime }}{G} ; \frac{1}{G}})$$ ( G ? ; 1 G ) ; expansion method ; The (2 $$+$$ + 1) ; dimensional nonlinear cubic–quintic Ginzburg–Landau equation ; Schr?dinger equations ; Exact traveling wave solutions ; Solitary wave solutions ; 35K99 ; 35P05 ; 35P99 ; 35C05
• 刊名：Ricerche di Matematica
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：64
• 期：1
• 页码：167-194
• 全文大小：1,334 KB
• 参考文献：1.Tian, B., Gao, Y.T., Zhu, H.W.: Variable-coefficient higher-order nonlinear Schr?dinger model in optical fibers: variable-coefficient bilinear form, B?cklund transformation, brightons and symbolic computation. Phys. Lett. A 366, 223-29 (2007)MATH View Article
  2.Nakazawa, M., Kubota, H., Suzuki, K., Yamada, E., Sahara, A.: Recent progress in soliton transmission technology. Chaos 10, 486-14 (2000)View Article
  3.Peacock, A.C., Kruhlak, R.J., Harvey, J.D., Dudley, J.M.: Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion. Opt. Commun. 206, 171-77 (2002)View Article
  4.Tian, B., Gao, Y.T.: Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas. Phys. Lett. A 362, 283-88 (2007)MATH View Article
  5.Hao, R., Li, L., Li, Z., Xue, W., Zhou, G.: A new approach to exact soliton solutions and soliton interaction for the nonlinear Schr?dinger equation with variable coefficients. Opt. Commun. 236, 79-6 (2004)View Article
  6.Gedalin, M., Scott, T.C., Band, Y.B.: Optical solitary waves in the higher order nonlinear Schr?dinger equation. Phys. Rev. Lett. 78, 448-51 (1997)View Article
  7.Mihalache, D., Torner, L., Moldoveanu, F., Panoiu, N.C., Truta, N.: Inverse scattering approach to femtosecond solitons in monomode optical fibers. Phys. Rev. E 48, 4699-709 (1993)View Article
  8.Shi, Y., Dai, Z., Li, D.: Application of Exp-function method for 2D cubic-quintic Ginzburg-Landau equation. Appl. Math. Comput. 210, 269-75 (2009)MATH MathSciNet View Article
  9.Triki, H., Hayat, T., Aldossary, O.M., Biswas, A.: Bright and dark solitons for the resonant nonlinear Schr?dinger’s equation with time-dependent coefficients. Opt. Laser Technol. 44, 2223-231 (2012)View Article
  10.Ablowitz, M.J., Clarkson, P.A.: Solitons. Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge University Press, New York (1991)MATH
  11.Hirota, R.: Exact solutions of the KdV equation for multiple collisions of solutions. Phys. Rev. Lett. 27, 1192-194 (1971)MATH View Article
  12.Weiss, J., Tabor, M., Carnevale, G.: The Painlev é property for partial differential equations. J. Math. Phys. 24, 522-26 (1983)MATH MathSciNet View Article
  13.Kudryashov, N.A.: On types of nonlinear nonintegrable equations with exact solutions. Phys. Lett. A 155, 269-75 (1991)MathSciNet View Article
  14.Rogers, C., Shadwick, W.F.: Backlund Transformations and Their Applications. Academic Press, New York (1982)
  15.Zhang, S.: Application of Exp-function method to high-dimensional nonlinear evolution equations. Chaos Solitons Fractals 38, 270-76 (2008)MATH MathSciNet View Article
  16.Bekir, A.: Application of the Exp-function method for nonlinear differential-difference equations. Appl. Math. Comput. 215, 4049-053 (2010)MATH MathSciNet View Article
  17.Abdou, M.A.: The extended tanh-method and its applications for solving nonlinear physical models. Appl. Math. Comput. 190, 988-96 (2007)MATH MathSciNet View Article
  18.Yusufoglu, E., Bekir, A.: Exact solutions of coupled nonlinear Klein-Gordon equations. Math. Comput. Model. 48, 1694-700 (2008)MATH MathSciNet View Article
  19.Zayed, E.M.E., Alurrfi, K.A.E.: The modified extended tanh-function method and its applications to the generalized KdV-mKdV equation with any-order nonlinear terms. Int. J. Environ. Eng. Sci. Technol. Res. 1(8), 165-70 (2013)
  20.Chen, Y., Wang, Q.: Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation. Chaos Solitons Fractals 24, 745-57 (2005)MATH MathSciNet View Article
  21.Lu, D.: Jacobi elliptic function solutions for two variant Boussinesq equations. Chaos Solitons Fractals 24, 1373-385 (2005)MathSciNet View Article
  22.Wang, M.L., Li, X., Zhang, J.: The \((\frac{G}{G}^{\prime })\) -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417-23 (2008)MATH MathSciNet View Article
  23.Zhang, S., Tong, J.L., Wang, W.: A generalized \((\frac{G}{G} ^{\prime })\) -expansion method for the mKdV equation with variable coefficients. Phys. Lett. A 372, 2254-257 (2008)MATH MathSciNet View Article
  24.Zayed, E.M.E., Gepreel, K.A.: The \((\frac{G}{G}^{\prime })\) -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys. 50, 013502-13512 (2009)MathSciNet View Article
  25.Zayed, E.M.E.: The \((\frac{G}{G}^{\prime })\) -expansion method and its applications to some nonlinear evolution equations in the mathematical physics. J. Appl. Math. Comput. 30, 89-03 (2009)MATH MathSciNet View Article
  26.Zayed, E.M.E.: Traveling wave solutions for higher dimensional nonlinear evolution equations using the \((\frac{G}{G}^{\prime })\) -expansion method. J. Appl. Math. Inform. 2
• 作者单位：E. M. E. Zayed (1)
  K. A. E. Alurrfi (1)
  
  1. Department of Mathematics, Faculty of Science, Zagazig University, P.O.Box 44519, Zagazig, Egypt
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebra
  Analysis
  Geometry
  Numerical Analysis
  Probability Theory and Stochastic Processes
  
• 出版者：Springer Milan
• ISSN：1827-3491

文摘

The propagation of the optical solitons is usually governed by the nonlinear Schr?dinger equations. In this article, the two variable \((\frac{ G^{\prime }}{G},\frac{1}{G})\)-expansion method is employed to construct the exact traveling wave solutions with parameters of two nonlinear PDEs namely, the (2\(+\)1)-dimensional nonlinear cubic–quintic Ginzburg–Landau equation and the (1\(+\)1)-dimensional resonant nonlinear Schr?dinger’s equation with dual-power law nonlinearity which describe the propagation of optical pulses in optic fibers. When the parameters are replaced by special values, the well-known solitary wave solutions of these equations rediscovered from the traveling waves. This method can be thought of as the generalization of well-known original \((\frac{G^{\prime }}{G})\)-expansion method proposed by M. Wang et al. It is shown that the two variable \((\frac{G^{\prime }}{G}, \frac{1}{G})\)-expansion method provides a more powerful mathematical tool for solving many other nonlinear PDEs in mathematical physics.