Exact Pfaffian and N-Soliton Solutions to a (3+1)-Dimensional Generalized Integrable Nonlinear Partial Differential Equations
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:51665", author = "Magdy G. Asaad", title = "Exact Pfaffian and N-Soliton Solutions to a (3+1)-Dimensional Generalized Integrable Nonlinear Partial Differential Equations", 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.", volume = "7", number = "4", pages = "575-8", }