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Exact Pfaffian and N-Soliton Solutions to a (3+1)-Dimensional Generalized Integrable Nonlinear Partial Differential Equations

Authors: Magdy G. Asaad

Abstract:

The objective of this paper is to use the Pfaffian technique to construct different classes of exact Pfaffian solutions and N-soliton solutions to some of the generalized integrable nonlinear partial differential equations in (3+1) dimensions. In this paper, I will show that the Pfaffian solutions to the nonlinear PDEs are nothing but Pfaffian identities. Solitons are among the most beneficial solutions for science and technology, from ocean waves to transmission of information through optical fibers or energy transport along protein molecules. The existence of multi-solitons, especially three-soliton solutions, is essential for information technology: it makes possible undisturbed simultaneous propagation of many pulses in both directions.

Keywords: Bilinear operator, G-BKP equation, Integrable nonlinear PDEs, Jimbo-Miwa equation, Ma-Fan equation, N-soliton solutions, Pfaffian solutions.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1057407

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[1] M. G. Asaad and W. X. Ma, "Pfaffian solution to a (3+1)-dimensional generalized BKP equation and its modified counterpart", Appl. Math. Comput., 2012, 218, 5524-5542.
[2] R. Hirota, "The Direct Method in Soliton Theory", Cambridge University Press, 2004.
[3] W. X. Ma, A. Abdeljabbar and M. G. Asaad, "Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation", Appl. Math. Comput., 2011, 217, 10016-10023.
[4] M. G. Asaad, W. X. Ma, "Extended Gram-type determinant, wave and rational solutions to two (3+1)-dimensional nonlinear evolution equations", Appl. Math. Comput., 2012, 219, 213-225.
[5] M. G. Asaad, "Pfaffian solutions to a (3+1)-dimensional Ma-Fan equation and its bilinear Backlund Transformation", to appear, 2013.
[6] M. G. Asaad, A. Abdeljabbar, "Application of the Pfaffian technique to two (3+1)-dimensional soliton equations of Jimbo-Miwa type", to appear, 2013.
[7] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, "A new hierarchy of soliton equations of KP-type", Physica D., 1981, 4, 343-365.
[8] B. Grammaticos, Y. Kosmann-Schwarzbach, T. Tamizhmani, "Discrete Integrable Systems", Springer, Berlin Heidelberg, 2004.
[9] R. E. Caianiello, "Combinatorics and Renormalization in Quantum Field Theory", New York: Benjamin, 1973.
[10] V. G. Kac, "Infinite Dimensinal Lie Algebras", Cambridge Univ. Press, New York, 1995.
[11] R. Hirota, "Soliton solutions to the BKP equations - I. The Pfaffian technique", J. Phys. Soc. Jpn., 1989, 58-7, 2285-2296.
[12] W. X. Ma, E. G. Fan, "Linear superposition principle applying to Hirota bilinear equations", Comput. Math. Appl., 2011, 61, 950-959.
[13] M. Jimbo, T. Miwa, "Solitonand infinite-dimensional Lie algebras", Publ. Res. Inst. Math. Soc. Kyoto Univ., 1983, 19, 943-1001.
[14] B. Dorrizzi, B. Grammaticos, A. Ramani, P. Winternitz, "Are all the equations of the KP hierarchy integrable?", J. Math. Phys., 1986, 27, 2848-2852.