Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm

Applied Mathematics and Computation

Volumn 218, Issue 24, 2012, Pages 11871-11879

  Ma, Wen Xiu ^a   Zhu, Zuonong ^b  

^a UNIVERSITY OF SOUTH FLORIDA   (United States)

^b SHANGHAI JIAO TONG UNIVERSITY   (China)

Author keywords

Hirota bilinear form; Multiple wave solution; Soliton equation

Indexed keywords

BILINEAR FORM; MULTIPLE WAVES; SOLITON EQUATION; WAVE FREQUENCIES;

PHASE SHIFT; SOLITONS;

ALGORITHMS;

EID: 84863780073     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2012.05.049     Document Type: Article

References (50)

1. R. Hirota Direct methods in soliton theory R.K. Bullough, P.J. Caudrey, Solitons 1980 Springer Berlin 157 176
2. R. Hirota The Direct Method in Soliton Theory 2004 Cambridge University Press New York
3. J. Hietarinta Hirota's bilinear method and soliton solutions Phys. AUC 15 Part 1 2005 31 37
4. W.X. Ma, T.W. Huang, and Y. Zhang A multiple exp-function method for nonlinear differential equations and its application Phys. Scri. 82 2010 065003 (8pp)
5. M.T. Darvishi1, M. Najafi, and M. Najafi Application of multiple exp-function method to obtain multi-soliton solutions of (2+1)- and (3+1)-dimensional breaking soliton equations Am. J. Comput. Appl. Math. 1 2011 41 47
6. W.X. Ma, and E.G. Fan Linear superposition principle applying to Hirota bilinear equations Comput. Math. Appl. 61 2011 950 959
7. W.X. Ma, Y. Zhang, Y.N. Tang, and J.Y. Tu Hirota bilinear equations with linear subspaces of solutions Appl. Math. Comput. 218 2012 7174 7183
8. R. Hirota, and M. Ito Resonance of solitons in one dimension J. Phys. Soc. Jpn. 52 1983 744 748
9. H.B. Lan, and K.L. Wang Exact solutions for two nonlinear equations I J. Phys. A Math. Gen. 23 1990 3923 3928
10. W. Malfliet, and W. Hereman The tanh method I: Exact solutions of nonlinear evolution and wave equations Phys. Scri. 54 1996 563 568
11. W.X. Ma, and B. Fuchssteiner Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation Int. J. Non-linear Mech. 31 1996 329 338
12. H.T. Chen, and H.C. Yin A note on the elliptic equation method Commun. Nonlinear Sci. Numer. Simulat. 13 2008 547 553
13. H.Q. Zhang Various exact travelling wave solutions for Kundu equation with fifth-order nonlinear term Rep. Math. Phys. 65 2010 231 239
14. Z.Y. Yan The new tri-function method to multiple exact solutions of nonlinear wave equations Phys. Scri. 78 2008 035001 (5pp)
15. G ′G) -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics Phys. Lett. A 372 2008 417 423
16. G ′ / G) -expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics J. Math. Phys. 50 2009 013502 (12pp)
17. W.X. Ma Complexiton solutions to the Korteweg-de Vries equation Phys. Lett. A 301 2002 35 44
18. W.X. Ma, and K. Maruno Complexiton solutions of the Toda lattice equation Phys. A 343 2004 219 237
19. W.X. Ma Soliton, positon and negaton solutions to a Schrödinger self-consistent source equation J. Phys. Soc. Jpn. 72 2003 3017 3019
20. W.X. Ma Complexiton solutions of the Korteweg-de Vries equation with self-consistent sources Chaos Solitons Fract. 26 2005 1453 1458
21. W.X. Ma, and Y. You Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions Trans. Am. Math. Soc. 357 2005 1753 1778
22. W.X. Ma, C.X. Li, and J.S. He A second Wronskian formulation of the Boussinesq equation Nonlinear Anal. 70 2009 4245 4258
23. C.X. Li, W.X. Ma, X.J. Liu, and Y.B. Zeng Wronskian solutions of the Boussinesq equation-solitons, negatons, positons and complexitons Inverse Probl. 23 2007 279 296
24. W.X. Ma An application of the Casoratian technique to the 2D Toda lattice equation Mod. Phys. Lett. B 22 2008 1815 1825
25. R. Hirota Soliton solutions to the BKP equations - I. The Pfaffian technique J. Phys. Soc. Jpn. 58 1989 2285 2296
26. G.F. Yu, and X.B. Hu Extended Gram-type determinant solutions to the Kadomtsev-Petviashvili equation Math. Comput. Simulat. 80 2009 184 191
27. N.A. Kudryashov Seven common errors in finding exact solutions of nonlinear differential equations Commun. Nonlinear Sci. Numer. Simulat. 14 2009 3507 3529
28. W.X. Ma Comment on the 3 + 1 dimensional Kadomtsev-Petviashvili equations Commun. Nonlinear Sci. Numer. Simulat. 16 2011 2663 2666
29. J.H. He, and H.X. Wu Exp-function method for nonlinear wave equations Chaos Solitons Fract. 30 2006 700 708
30. S.A. El-Wakil, M.A. Madkour, and M.A. Abdou Application of Exp-function method for nonlinear evolution equations with variable coefficients Phys. Lett. A 369 2007 62 69
31. F. Khanib, S. Hamedi-Nezhad, M.T. Darvishi, and S.-W. Ryua New solitary wave and periodic solutions of the foam drainage equation using the exp-function method Nonlinear Anal. Real World Appl. 10 2009 1904 1911
32. W.X. Ma Darboux transformations for a Lax integrable system in 2 n dimensions Lett. Math. Phys. 39 1997 33 49
33. H.Q. Zhao, Z.N. Zhu, Darboux transformation and exact solutions for a three-field lattice equation, in: Nonlinear and Modern Mathematical Physics, AIP Conf. Proc. Vol. 1212, Am. Inst. Phys., Melville, NY, 2010, pp. 162-169.
34. W. Hereman, and A. Nuseir Symbolic methods to construct exact solutions of nonlinear partial differential equations Math. Comput. Simulat. 43 1997 13 27
35. A.M. Wazwaz N-soliton solutions for shallow water waves equations in (1+1) and (2+1) dimensions Appl. Math. Comput. 217 2011 8840 8845
36. W.X. Ma, and A. Pekcan Uniqueness of the Kadomtsev-Petviashvili and Boussinesq equations Z. Naturforsch. A 66 2011 282 377
37. Z.N. Zhu Painlevé property, Bäcklund transformation, Lax pair and soliton-like solution for a variable coefficient KP equation Phys. Lett. A 182 1993 277 281
38. F.C. You, T.C. Xia, and D.Y. Chen Decomposition of the generalized KP, cKP and mKP and their exact solutions Phys. Lett. A 372 2008 3184 3194
39. A.M. Wazwaz Four (2 + 1)-dimensional integrable extensions of the Kadomtsev-Petviashvili equation Appl. Math. Comput. 215 2010 3631 3644
40. R. Hirota A new form of Bäcklund transformations and its relation to the inverse scattering problem Prog. Theor. Phys. 52 1974 1498 1512
41. C. Gilson, F. Lambert, J. Nimmo, and R. Willox On the combinatorics of the Hirota DD-operators Proc. Roy. Soc. London Ser. A 452 1996 223 234
42. W.X. Ma A refined invariant subspace method and applications to evolution equations Sci. China Math. 2012
43. A.M. Wazwaz Multiple-soliton solutions for a (3 + 1) -dimensional generalized KP equation Commun. Nonlinear Sci. Numer. Simulat. 17 2012 491 495
44. W.X. Ma, A. Abdeljabbar, and M.G. Asaad Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation Appl. Math. Comput. 217 2001 10016 10023
45. W.X. Ma, T.C. Xia, Pfaffianized systems for a generalized Kadomtsev-Petviashvili equation, Preprint, 2012.
46. X.G. Geng, and Y.L. Ma N-soliton solution and its Wronskian form of a (3 + 1) -dimensional nonlinear evolution equation Phys. Lett. A 369 2007 285 289
47. W.X. Ma, and J.-H. Lee A transformed rational function method and exact solutions to the 3 + 1 dimensional Jimbo-Miwa equation Chaos Solitons Fract. 42 2009 1356 1363
48. J.P. Wu, and X.G. Geng Grammian determinant solution and Pfaffianization for a (3 + 1) -dimensional soliton equation Commun. Theor. Phys. 52 2009 791 794
49. X.B. Hu, and H.Y. Wang Construction of dKP and BKP equations with self-consistent sources Inverse Probl. 22 2006 1903 1920
50. W.X. Ma Generalized bilinear differential equations World Appl. Sci. J. 2012