The (G'/G)-expansion method and its applications to some nonlinear evolution equations in the mathematical physics

Journal of Applied Mathematics and Computing

Volumn 30, Issue 1-2, 2009, Pages 89-103

  Zayed, E M E ^a,b  

^a Zagazig University   (Egypt)

^b TAIF UNIVERSITY   (Saudi Arabia)

Author keywords

The (G' G) expansion method; The Boiti Leon Pempinelli equations; The dispersive long wave equations; The Nizhnik Novikov Vesselov equations; The Painlev integrable Burgers equations; Traveling wave solutions

Indexed keywords

EID: 63949088141     PISSN: 15985865     EISSN: None     Source Type: Journal    
DOI: 10.1007/s12190-008-0159-8     Document Type: Article

References (41)

1. 1. M.A. Abdou 2007 The extended tanh-method and its applications for solving nonlinear physical models Appl. Math. Comput. 190 988 996 1123.65103 10.1016/j.amc.2007.01.070 2338770 Abdou, M.A.: The extended tanh-method and its applications for solving nonlinear physical models. Appl. Math. Comput. 190, 988–996 (2007)
2. 2. M.A. Abdou 2007 The extended F-expansion method and its applications for a class of nonlinear evolution equation Chaos Solitons Fractals 31 95 104 1138.35385 10.1016/j.chaos.2005.09.030 2263268 Abdou, M.A.: The extended F-expansion method and its applications for a class of nonlinear evolution equation. Chaos Solitons Fractals 31, 95–104 (2007)
3. 3. M.J. Ablowitz P.A. Clarkson 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform Cambridge University Press Cambridge Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform. Cambridge University Press, Cambridge (1991)
4. 4. A. Bekir 2008 Application of the ( $\\frac{G'}{G})$-expansion method for nonlinear evolution equations Phys. Lett. A 372 3400 3406 10.1016/j.physleta.2008.01.057 2414708 Bekir, A.: Application of the ( $\\frac{G'}{G})$-expansion method for nonlinear evolution equations. Phys. Lett. A 372, 3400–3406 (2008)
5. 5. A. Bekir A. Boz 2008 Exact solutions for nonlinear evolution equations using Exp-function method Phys. Lett. A 372 1619 1625 10.1016/j.physleta.2007.10.018 2396241 Bekir, A., Boz, A.: Exact solutions for nonlinear evolution equations using Exp-function method. Phys. Lett. A 372, 1619–1625 (2008)
6. 6. M. Boiti J. Leon P. Pempinelli 1987 Spectral transform for a two spatial dimension extension of the dispersive long wave equation Inverse Probl. 3 371 387 0641.35067 10.1088/0266-5611/3/3/007 909404 Boiti, M., Leon, J., Pempinelli, P.: Spectral transform for a two spatial dimension extension of the dispersive long wave equation. Inverse Probl. 3, 371–387 (1987)
7. 7. Y. Chen Q. Wang 2005 Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation Chaos Solitons Fractals 24 745 757 1072.35509 10.1016/j.chaos.2004.09.014 2116283 Chen, Y., Wang, Q.: Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation. Chaos Solitons Fractals 24, 745–757 (2005)
8. 8. E.G. Fan 2000 Extended tanh-function method and its applications to nonlinear equations Phys. Lett. A 277 212 218 05188425 10.1016/S0375-9601(00)00725-8 1827770 Fan, E.G.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)
9. 9. J.H. He X.H. Wu 2006 Exp-function method for nonlinear wave equations Chaos Solitons Fractals 30 700 708 1141.35448 10.1016/j.chaos.2006.03.020 2238695 He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30, 700–708 (2006)
10. 10. K.Z. Hong B. Wu X.F. Chen 2003 Painlevé analysis and some solutions of (2+1)-dimensional generalized Burgers equations Commun. Theor. Phys. 39 393 394 2001122 Hong, K.Z., Wu, B., Chen, X.F.: Painlevé analysis and some solutions of (2+1)-dimensional generalized Burgers equations. Commun. Theor. Phys. 39, 393–394 (2003)
11. 11. D.J. Huang H.Q. Zhang 2004 Exact traveling wave solutions for the Boiti-Leon-Pempinelli equation Chaos Solitons Fractals 22 243 247 1067.35085 10.1016/j.chaos.2004.01.004 2058766 Huang, D.J., Zhang, H.Q.: Exact traveling wave solutions for the Boiti-Leon-Pempinelli equation. Chaos Solitons Fractals 22, 243–247 (2004)
12. 12. S. Liu Z. Fu S.D. Liu Q. Zhao 2001 Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations Phys. Lett. A 289 69 74 0972.35062 10.1016/S0375-9601(01)00580-1 1862082 Liu, S., Fu, Z., Liu, S.D., Zhao, Q.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289, 69–74 (2001)
13. 13. X.Z. Li M.L. Wang 2007 A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with higher order nonlinear terms Phys. Lett. A 361 115 118 05200759 10.1016/j.physleta.2006.09.022 2287764 Li, X.Z., Wang, M.L.: A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with higher order nonlinear terms. Phys. Lett. A 361, 115–118 (2007)
14. 14. Z. Lü H. Zhang 2004 Soliton-like and multi-soliton like solutions of the Boiti-Leon-Pempinelli equation Chaos Solitons Fractals 19 527 531 1068.35137 10.1016/S0960-0779(03)00104-8 2012989 Lü, Z., Zhang, H.: Soliton-like and multi-soliton like solutions of the Boiti-Leon-Pempinelli equation. Chaos Solitons Fractals 19, 527–531 (2004)
15. 15. M.R. Miura 1978 Backlund Transformation Springer Berlin Miura, M.R.: Backlund Transformation. Springer, Berlin (1978)
16. 16. Y. Ren H. Zhang 2006 New generalized hyperbolic functions and auto-Backlund transformation to find new exact solutions of the (2+1)-dimensional NNY equation Phys. Lett. A 357 438 448 05358672 10.1016/j.physleta.2006.04.082 2244546 Ren, Y., Zhang, H.: New generalized hyperbolic functions and auto-Backlund transformation to find new exact solutions of the (2+1)-dimensional NNY equation. Phys. Lett. A 357, 438–448 (2006)
17. 17. C. Rogers W.F. Shadwick 1982 Backlund Transformations Academic Press New York Rogers, C., Shadwick, W.F.: Backlund Transformations. Academic Press, New York (1982)
18. 18. Z. Wang H.Q. Zhang 2007 A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equation with nonlinear terms of any order Appl. Math. Comput. 186 693 704 1115.35116 10.1016/j.amc.2006.08.015 2314530 Wang, Z., Zhang, H.Q.: A new generalized Riccati equation rational expansion method to a class of nonlinear evolution equation with nonlinear terms of any order. Appl. Math. Comput. 186, 693–704 (2007)
19. 19. M. Wang Y. Zhou 2003 The periodic wave equations for the Klein-Gordon-Schrodinger equations Phys. Lett. A 318 84 92 1098.81770 10.1016/j.physleta.2003.07.026 2020906 Wang, M., Zhou, Y.: The periodic wave equations for the Klein-Gordon-Schrodinger equations. Phys. Lett. A 318, 84–92 (2003)
20. 20. M. Wang X. Li 2005 Extended F-expansion and periodic wave solutions for the generalized Zakharov equations Phys. Lett. A 343 48 54 05314059 10.1016/j.physleta.2005.05.085 2149694 Wang, M., Li, X.: Extended F-expansion and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A 343, 48–54 (2005)
21. 21. M. Wang X. Li 2005 Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation Chaos Solitons Fractals 24 1257 1268 1092.37054 10.1016/j.chaos.2004.09.044 2123272 Wang, M., Li, X.: Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos Solitons Fractals 24, 1257–1268 (2005)
22. 22. M.L. Wang X.Z. Li J.L. Zhang 2007 Sub-ODE method and solitary wave solutions for higher order nonlinear Schrodinger equation Phys. Lett. A 363 96 101 10.1016/j.physleta.2006.10.077 2287805 Wang, M.L., Li, X.Z., Zhang, J.L.: Sub-ODE method and solitary wave solutions for higher order nonlinear Schrodinger equation. Phys. Lett. A 363, 96–101 (2007)
23. 23. D.S. Wang Y.J. Ren H.Q. Zhang 2005 Further extended sinh-cosh and sin-cos methods and new non traveling wave solutions of the (2+1)-dimensional dispersive long wave equations Appl. Math. E Notes 5 157 163 1091.35072 2133116 Wang, D.S., Ren, Y.J., Zhang, H.Q.: Further extended sinh-cosh and sin-cos methods and new non traveling wave solutions of the (2+1)-dimensional dispersive long wave equations. Appl. Math. E Notes 5, 157–163 (2005)
24. 24. M. Wang X. Li J. Zhang 2008 The ( $\\frac{G'}{G})$-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics Phys. Lett. A 372 417 423 10.1016/j.physleta.2007.07.051 2381823 Wang, M., Li, X., Zhang, J.: The ( $\\frac{G'}{G})$-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417–423 (2008)
25. 25. Z. Yan H. Zhang 2001 New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water Phys. Lett. A 285 355 362 0969.76518 10.1016/S0375-9601(01)00376-0 1851579 Yan, Z., Zhang, H.: New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water. Phys. Lett. A 285, 355–362 (2001)
26. 26. Z. Yan 2003 Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey-Stawartson-type equation via a new method Chaos Solitons Fractals 18 299 309 1069.37060 10.1016/S0960-0779(02)00653-7 1983423 Yan, Z.: Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey-Stawartson-type equation via a new method. Chaos Solitons Fractals 18, 299–309 (2003)
27. 27. E. Yomba 2008 Construction of new soliton-like solutions of the (2+1)-dimensional dispersive long wave equation Chaos Solitons Fractals 20 1135 1139 10.1016/j.chaos.2003.09.026 2030345 Yomba, E.: Construction of new soliton-like solutions of the (2+1)-dimensional dispersive long wave equation. Chaos Solitons Fractals 20, 1135–1139 (2008)
28. 28. E. Yusufoglu 2008 New solitary solutions for the MBBM equations using Exp-function method Phys. Lett. A 372 442 446 10.1016/j.physleta.2007.07.062 2418598 Yusufoglu, E.: New solitary solutions for the MBBM equations using Exp-function method. Phys. Lett. A 372, 442–446 (2008)
29. 29. E. Yusufoglu A. Bekir 2008 Exact solution of coupled nonlinear evolution equations Chaos Solitons Fractals 37 842 848 1148.35352 10.1016/j.chaos.2006.09.074 2400334 Yusufoglu, E., Bekir, A.: Exact solution of coupled nonlinear evolution equations. Chaos Solitons Fractals 37, 842–848 (2008)
30. 30. E.M.E. Zayed H.A. Zedan K.A. Gepreel 2004 On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV equations Chaos Solitons Fractals 22 285 303 1069.35080 10.1016/j.chaos.2003.12.045 2060865 Zayed, E.M.E., Zedan, H.A., Gepreel, K.A.: On the solitary wave solutions for nonlinear Hirota-Satsuma coupled KdV equations. Chaos Solitons Fractals 22, 285–303 (2004)
31. 31. E.M.E. Zayed H.A. Zedan K.A. Gepreel 2004 On the solitary wave solutions for nonlinear Euler equations Appl. Anal. 83 1101 1132 1061.35072 10.1080/00036810410001689274 2103297 Zayed, E.M.E., Zedan, H.A., Gepreel, K.A.: On the solitary wave solutions for nonlinear Euler equations. Appl. Anal. 83, 1101–1132 (2004)
32. 32. E.M.E. Zayed H.A. Zedan K.A. Gepreel 2004 Group analysis and modified tanh-function to find the invariant solutions and soliton solution for nonlinear Euler equations Int. J. Nonlinear Sci. Numer. Simul. 5 221 234 Zayed, E.M.E., Zedan, H.A., Gepreel, K.A.: Group analysis and modified tanh-function to find the invariant solutions and soliton solution for nonlinear Euler equations. Int. J. Nonlinear Sci. Numer. Simul. 5, 221–234 (2004)
33. 33. E.M.E. Zayed A.M. Abourabia K.A. Gepreel M.M. Horbaty 2007 Traveling solitary wave solutions for the nonlinear coupled KdV system Chaos Solitons Fractals 34 292 306 1151.35422 10.1016/j.chaos.2006.03.065 2327408 Zayed, E.M.E., Abourabia, A.M., Gepreel, K.A., Horbaty, M.M.: Traveling solitary wave solutions for the nonlinear coupled KdV system. Chaos Solitons Fractals 34, 292–306 (2007)
34. 34. E.M.E. Zayed K.A. Gepreel M.M. Horbaty 2008 Exact solutions for some nonlinear differential equations using complex hyperbolic function Appl. Anal. 87 509 522 05309435 10.1080/00036810801912098 2418838 Zayed, E.M.E., Gepreel, K.A., Horbaty, M.M.: Exact solutions for some nonlinear differential equations using complex hyperbolic function. Appl. Anal. 87, 509–522 (2008)
35. 35. S.L. Zhang B. Wu S.Y. Lou 2002 Painlevé analysis and special solutions of generalized Broer-Kaup equations Phys. Lett. A 300 40 48 0997.76012 10.1016/S0375-9601(02)00688-6 1927441 Zhang, S.L., Wu, B., Lou, S.Y.: Painlevé analysis and special solutions of generalized Broer-Kaup equations. Phys. Lett. A 300, 40–48 (2002)
36. 36. S. Zhang T.C. Xia 2006 A generalized F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations Appl. Math. Comput. 183 1190 1200 1111.35318 10.1016/j.amc.2006.06.043 2294076 Zhang, S., Xia, T.C.: A generalized F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equations. Appl. Math. Comput. 183, 1190–1200 (2006)
37. 37. S. Zhang 2008 Application of Exp-function method to higher dimensional nonlinear evolution equation Chaos Solitons Fractals 38 270 276 1142.35593 10.1016/j.chaos.2006.11.014 2417662 Zhang, S.: Application of Exp-function method to higher dimensional nonlinear evolution equation. Chaos Solitons Fractals 38, 270–276 (2008)
38. 38. S. Zhang 2008 Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations Phys. Lett. A 372 1873 1880 10.1016/j.physleta.2007.10.086 2398392 Zhang, S.: Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer-Kaup-Kupershmidt equations. Phys. Lett. A 372, 1873–1880 (2008)
39. 39. S. Zhang T.C. Xia 2008 A further improved tanh-function method exactly solving the (2+1)-dimensional dispersive long wave equations Appl. Math. E Notes 8 58 66 05497778 2383325 Zhang, S., Xia, T.C.: A further improved tanh-function method exactly solving the (2+1)-dimensional dispersive long wave equations. Appl. Math. E Notes 8, 58–66 (2008)
40. 40. S. Zhang J. Tong W. Wang 2008 A generalized ( $\\frac{G'}{G}$)-expansion method for the mKdV equation with variable coefficients Phys. Lett. A 372 2254 2257 10.1016/j.physleta.2007.11.026 2418595 Zhang, S., Tong, J., Wang, W.: A generalized ( $\\frac{G'}{G}$)-expansion method for the mKdV equation with variable coefficients. Phys. Lett. A 372, 2254–2257 (2008)
41. 41. J. Zhang X. Wei Y. Lu 2008 A generalized ( $\\frac{G'}{G}$)-expansion method and its applications Phys. Lett. A 372 3653 3658 10.1016/j.physleta.2008.02.027 2413864 Zhang, J., Wei, X., Lu, Y.: A generalized ( $\\frac{G'}{G}$)-expansion method and its applications. Phys. Lett. A 372, 3653–3658 (2008)