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Year 2018, Volume: 1 Issue: 4, 273 - 279, 20.12.2018
https://doi.org/10.32323/ujma.399596

Abstract

References

  • [1] L. Debtnath, Nonlinear partial differential equations for scientist and engineers, Birkhauser, Boston, MA, 1997.
  • [2] A. M. Wazwaz, Partial differential equations: methods and applications, Balkema, Rotterdam, 2002.
  • [3] Y. Shang, Backlund transformation,Lax pairs and explicit exact solutions for the shallow water wave sequation, Appl. Math. Comput., 187 (2007), 1286-1297.
  • [4] T. L. Bock, M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-Onoequation, Phys. Lett. A, 74 (1979), 173-176.
  • [5] V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Springer, Berlin, 1991.
  • [6] A.M. Abourabia, M. M. El Horbaty, On solitary wave solutions for the two-dimensional nonlinear modified Kortweg-de Vries-Burger equation, Chaos Solitons Fractals, 29 (2006), 354-364.
  • [7] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.
  • [8] Y. Chuntao, A simple transformation for nonlinear waves, Phys. Lett. A, 224 (1996), 77-84.
  • [9] F. Cariello, M. Tabor, Painlev eexpansions for nonintegrable evolution equations, Phys. D, 39(1989), 77-94.
  • [10] E. Fan, Two new application of the homogeneous balance method, Phys. Lett. A, 265 (2000), 353-357.
  • [11] P. A. Clarkson, New similarity solutions for the modified boussinesq equation, J. Phys. A: Math. Gen., 22 (1989), 2355-2367.
  • [12] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.
  • [13] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218.
  • [14] S. A. Elwakil, S. K. El-labany, M. A. Zahran, R. Sabry, Modified extended tanh- function method for solving nonlinear partial differential equations, Phys. Lett. A, 299 (2002), 179-188.
  • [15] H. Chen, H. Zhang, New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos Solitons Fractals, 19 (2004), 71-76.
  • [16] Z. Fu, S. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290 (2001), 72-76.
  • [17] S. Shen, Z. Pan, A note on the Jacobi elliptic function expansion method, Phys. Let. A, 308 (2003), 143-148.
  • [18] H. T. Chen, Z. Hong-Qing, New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation, Chaos Solitons Fractals, 20 (2004), 765-769.
  • [19] Y. Chen, Q. Wang, B. Li, Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly periodic solutions of nonlinear evolution equations, Z. Naturforsch. A, 59 (2004), 529-536.
  • [20] Y. Chen, Z. Yan, The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos Solitons Fractals, 29 (2006), 948-964.
  • [21] M. Wang, X. Li, J. Zhang, The $\left( {\frac{{G'}}{G}} \right)$-expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A, 372 (2008), 417-423.
  • [22] S. Guo, Y. Zhou, The extended $\left( {\frac{{G'}}{G}} \right)$-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Appl. Math. Comput., 215 (2010), 3214-3221.
  • [23] H. L. Lu, X. Q. Liu, L. Niu, A generalized $\left( {\frac{{G'}}{G}} \right)$-expansion method and its applications to nonlinear evolution equations, Appl. Math. Comput., 215 (2010), 3811-3816.
  • [24] L. Li, E. Li, M. Wang, The $\left( {\frac{{G'}}{G},\frac{1}{G}} \right)$ -expansion method and its application to travelling wave solutions of the Zakharov equations, Appl. Math-A J. Chin. U., 25 (2010), 454-462.
  • [25] J. Manafian, Optical soliton solutions for Schrodinger type nonlinear evolution equations by the tan $\left( {\frac{{\phi \left( \varphi \right)}}{2}} \right)$- expansion Method, Optik, 127 (2016), 4222-4245.
  • [26] E. Don, Schaum’s outline of Theoryand problems of mathematica, McGraw-Hill, 2001.
  • [27] E. Yasar, New travelling wave solutions to the Ostrovsky equation, Appl. Math. Comput., 216(11) (2010), 3191-3194.
  • [28] Y. Yildirim, E. Yasar, An extended Korteweg-de Vries equation: multi-soliton solutions and conservation laws, Nonlinear Dynam., 90(3) (2017), 1571-1579.
  • [29] E. Yasar, S. San, Y. S. Ozkan, Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation, Open Phys., 14(1) (2016), 37-43.
  • [30] W. X. Ma, Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations, 264(4) (2018), 2633-2659.
  • [31] W. X. Ma, T. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Physica Scr., 82(6) (2010), 065003.
  • [32] M. S. Osman, H. I. Abdel-Gawad, M. A. El Mahdy, Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion, Results Phys., 8 (2018), 1054-1060.
  • [33] Hamdy I. Abdel-Gawad, M. S. Osman, On the variational approach for analyzing the stability of solutions of evolution equations, Kyungpook Math., 53 (2013), 661-680.
  • [34] M. S. Osman, On complex wave solutions governed by the 2D Ginzburg-Landau equation with variable coefficients, Optik, 156 (2018), 169-174.
  • [35] M. S. Osman, J. A. T. Machado, D. Baleanu, On nonautonomous complex wave solutions described by the coupled Schrodinger-Boussinesq equation with variable-coefficients, Opt. Quant. Electron, 50 (2018), 73.
  • [36] M. Nasir Ali, M. S. Osman, S. Muhammad Husnine, On the analytical solutions of conformable time-fractional extended Zakharov-Kuznetsov equation through $\left( {\frac{{G'}}{{G2}}} \right)$-expansion method and the modified Kudryashov method, SeMA Journal, (in Press).
  • [37] M. S. Osman, Multiwave solutions of time-fractional (2 + 1)-dimensional Nizhnik-Novikov-Veselov equations, Pramana J. Phys., 88 (2017), 67.
  • [38] M. S. Osman, A. Majid Wazwaz, An efficient algorithm to construct multi-soliton rational solutions of the (2+ 1)-dimensional KdV equation with variable coefficients, Appl. Math. Comput., 321 (2018), 282-289.
  • [39] M. S. Osman, On multi-soliton solutions for the (2 + 1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide, Comput. Math. Appl., 75 (2018), 1-6.

Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations

Year 2018, Volume: 1 Issue: 4, 273 - 279, 20.12.2018
https://doi.org/10.32323/ujma.399596

Abstract

In this paper, we implemented an improved tanh function Method for multiple soliton solutions of new coupled Konno-Oono equation and extended (3+1)-dimensional KdV-type equation.

References

  • [1] L. Debtnath, Nonlinear partial differential equations for scientist and engineers, Birkhauser, Boston, MA, 1997.
  • [2] A. M. Wazwaz, Partial differential equations: methods and applications, Balkema, Rotterdam, 2002.
  • [3] Y. Shang, Backlund transformation,Lax pairs and explicit exact solutions for the shallow water wave sequation, Appl. Math. Comput., 187 (2007), 1286-1297.
  • [4] T. L. Bock, M. D. Kruskal, A two-parameter Miura transformation of the Benjamin-Onoequation, Phys. Lett. A, 74 (1979), 173-176.
  • [5] V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Springer, Berlin, 1991.
  • [6] A.M. Abourabia, M. M. El Horbaty, On solitary wave solutions for the two-dimensional nonlinear modified Kortweg-de Vries-Burger equation, Chaos Solitons Fractals, 29 (2006), 354-364.
  • [7] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.
  • [8] Y. Chuntao, A simple transformation for nonlinear waves, Phys. Lett. A, 224 (1996), 77-84.
  • [9] F. Cariello, M. Tabor, Painlev eexpansions for nonintegrable evolution equations, Phys. D, 39(1989), 77-94.
  • [10] E. Fan, Two new application of the homogeneous balance method, Phys. Lett. A, 265 (2000), 353-357.
  • [11] P. A. Clarkson, New similarity solutions for the modified boussinesq equation, J. Phys. A: Math. Gen., 22 (1989), 2355-2367.
  • [12] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys., 60 (1992), 650-654.
  • [13] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212-218.
  • [14] S. A. Elwakil, S. K. El-labany, M. A. Zahran, R. Sabry, Modified extended tanh- function method for solving nonlinear partial differential equations, Phys. Lett. A, 299 (2002), 179-188.
  • [15] H. Chen, H. Zhang, New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos Solitons Fractals, 19 (2004), 71-76.
  • [16] Z. Fu, S. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A, 290 (2001), 72-76.
  • [17] S. Shen, Z. Pan, A note on the Jacobi elliptic function expansion method, Phys. Let. A, 308 (2003), 143-148.
  • [18] H. T. Chen, Z. Hong-Qing, New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation, Chaos Solitons Fractals, 20 (2004), 765-769.
  • [19] Y. Chen, Q. Wang, B. Li, Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly periodic solutions of nonlinear evolution equations, Z. Naturforsch. A, 59 (2004), 529-536.
  • [20] Y. Chen, Z. Yan, The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos Solitons Fractals, 29 (2006), 948-964.
  • [21] M. Wang, X. Li, J. Zhang, The $\left( {\frac{{G'}}{G}} \right)$-expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A, 372 (2008), 417-423.
  • [22] S. Guo, Y. Zhou, The extended $\left( {\frac{{G'}}{G}} \right)$-expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Appl. Math. Comput., 215 (2010), 3214-3221.
  • [23] H. L. Lu, X. Q. Liu, L. Niu, A generalized $\left( {\frac{{G'}}{G}} \right)$-expansion method and its applications to nonlinear evolution equations, Appl. Math. Comput., 215 (2010), 3811-3816.
  • [24] L. Li, E. Li, M. Wang, The $\left( {\frac{{G'}}{G},\frac{1}{G}} \right)$ -expansion method and its application to travelling wave solutions of the Zakharov equations, Appl. Math-A J. Chin. U., 25 (2010), 454-462.
  • [25] J. Manafian, Optical soliton solutions for Schrodinger type nonlinear evolution equations by the tan $\left( {\frac{{\phi \left( \varphi \right)}}{2}} \right)$- expansion Method, Optik, 127 (2016), 4222-4245.
  • [26] E. Don, Schaum’s outline of Theoryand problems of mathematica, McGraw-Hill, 2001.
  • [27] E. Yasar, New travelling wave solutions to the Ostrovsky equation, Appl. Math. Comput., 216(11) (2010), 3191-3194.
  • [28] Y. Yildirim, E. Yasar, An extended Korteweg-de Vries equation: multi-soliton solutions and conservation laws, Nonlinear Dynam., 90(3) (2017), 1571-1579.
  • [29] E. Yasar, S. San, Y. S. Ozkan, Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation, Open Phys., 14(1) (2016), 37-43.
  • [30] W. X. Ma, Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations, 264(4) (2018), 2633-2659.
  • [31] W. X. Ma, T. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Physica Scr., 82(6) (2010), 065003.
  • [32] M. S. Osman, H. I. Abdel-Gawad, M. A. El Mahdy, Two-layer-atmospheric blocking in a medium with high nonlinearity and lateral dispersion, Results Phys., 8 (2018), 1054-1060.
  • [33] Hamdy I. Abdel-Gawad, M. S. Osman, On the variational approach for analyzing the stability of solutions of evolution equations, Kyungpook Math., 53 (2013), 661-680.
  • [34] M. S. Osman, On complex wave solutions governed by the 2D Ginzburg-Landau equation with variable coefficients, Optik, 156 (2018), 169-174.
  • [35] M. S. Osman, J. A. T. Machado, D. Baleanu, On nonautonomous complex wave solutions described by the coupled Schrodinger-Boussinesq equation with variable-coefficients, Opt. Quant. Electron, 50 (2018), 73.
  • [36] M. Nasir Ali, M. S. Osman, S. Muhammad Husnine, On the analytical solutions of conformable time-fractional extended Zakharov-Kuznetsov equation through $\left( {\frac{{G'}}{{G2}}} \right)$-expansion method and the modified Kudryashov method, SeMA Journal, (in Press).
  • [37] M. S. Osman, Multiwave solutions of time-fractional (2 + 1)-dimensional Nizhnik-Novikov-Veselov equations, Pramana J. Phys., 88 (2017), 67.
  • [38] M. S. Osman, A. Majid Wazwaz, An efficient algorithm to construct multi-soliton rational solutions of the (2+ 1)-dimensional KdV equation with variable coefficients, Appl. Math. Comput., 321 (2018), 282-289.
  • [39] M. S. Osman, On multi-soliton solutions for the (2 + 1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide, Comput. Math. Appl., 75 (2018), 1-6.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

İbrahim Enam İnan

Publication Date December 20, 2018
Submission Date February 28, 2018
Acceptance Date May 7, 2018
Published in Issue Year 2018 Volume: 1 Issue: 4

Cite

APA İnan, İ. E. (2018). Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations. Universal Journal of Mathematics and Applications, 1(4), 273-279. https://doi.org/10.32323/ujma.399596
AMA İnan İE. Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations. Univ. J. Math. Appl. December 2018;1(4):273-279. doi:10.32323/ujma.399596
Chicago İnan, İbrahim Enam. “Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations”. Universal Journal of Mathematics and Applications 1, no. 4 (December 2018): 273-79. https://doi.org/10.32323/ujma.399596.
EndNote İnan İE (December 1, 2018) Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations. Universal Journal of Mathematics and Applications 1 4 273–279.
IEEE İ. E. İnan, “Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations”, Univ. J. Math. Appl., vol. 1, no. 4, pp. 273–279, 2018, doi: 10.32323/ujma.399596.
ISNAD İnan, İbrahim Enam. “Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations”. Universal Journal of Mathematics and Applications 1/4 (December 2018), 273-279. https://doi.org/10.32323/ujma.399596.
JAMA İnan İE. Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations. Univ. J. Math. Appl. 2018;1:273–279.
MLA İnan, İbrahim Enam. “Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations”. Universal Journal of Mathematics and Applications, vol. 1, no. 4, 2018, pp. 273-9, doi:10.32323/ujma.399596.
Vancouver İnan İE. Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations. Univ. J. Math. Appl. 2018;1(4):273-9.

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