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Generalized Wronskian and Grammian solutions to a isospectral B-type Kadomtsev-Petviashvili equation. (English) Zbl 1421.76043
Summary: Generally speaking, the BKP hierarchy which only has Pfaffian solutions. In this paper, based on the Grammian and Wronskian derivative formulae, generalized Wronskian and Grammian determinant solutions are obtained for the isospectral BKP equation (the second member on the BKP hierarchy) in the Hirota bilinear form. Especially, with the help of the properties of the computing of Young diagram, we have first applied Young diagram proved the proposition of this paper. Moreover, by considering the different combinations of the entries in Wronskian, we obtain various types of Wronskian solutions.
76B25 Solitary waves for incompressible inviscid fluids
35G20 Nonlinear higher-order PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
Full Text: DOI
[1] Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform, (1981), SIAM: SIAM, Philadelphia
[2] Bullough, R. K.; Caudrey, P. J., Solitons, (1980), Springer-Verlag: Springer-Verlag, Berlin
[3] Benny, D. J., J Math Phys, 45, 52, (1966)
[4] Barriola, M.; Vilenkin, A., Phys Rev Lett, 63, 341, (1989)
[5] Colin, T., Physica D, 64, 215, (1993)
[6] Chen, D. Y.; Xin, H. W.; Zhang, D. J., Chaos, Solitons and Fractals, 15, 761, (2003)
[7] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., Phys.D, 82, 343, (1981)
[8] Date, E.; Jimbo, M.; Kashiwara, M.; Miwa, T., J.Phys.Soc.Jpn, 50, 3183, (1981)
[9] Freeman, N. C.; Nimmo, J. J.C., Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The Wronskian technique, Physics.Letters.A., 95, 1-3, (1983) · Zbl 0588.35077
[10] Hong, W. P., Phys Lett A, 361, 520, (2007)
[11] Hirota, R., Phys.Rev.Lett., 27, 1192, (1971)
[12] Hirota, R., The direct method in soliton theory, (2004), Cambridge: Cambridge University Press, Cambridge
[13] Hirota, R., Tang, Solutions of the Classical Boussinesq Equation and the Spherical Boussinesq Equation:The Wronskian Technique, J.Phys.Soc.Jpn., 55, 2137-2150, (1986)
[14] Hirota, R., J.Phys.Soc.Jpn, 58, 2285, (1989)
[15] Jimbo, M.; Miwa, T., Solitons and infinite dimensional Lie algebra, Res.I Math.Sci., 19, 943-1001, (1983) · Zbl 0557.35091
[16] Kasch, F., Modules and Rings, (1982), Academic Press: Academic Press, London, New York · Zbl 0523.16001
[17] Matveev, V. B.; Salle, M. A., Darboux Transformations and Solitons, (1991), Springer: Springer, Berlin · Zbl 0744.35045
[18] Nimmo, J. J.C.; Freeman, N. C., Physics.Letters., 96A, 443-446, (1983)
[19] Radhakrishnan, R.; Lakshmanan, M., Hietarinta J. Phys Rev E, 56, 2213, (1997)
[20] Rogers, C.; Shadwick, W. R., Bäcklund transformation and their application, (1982), Academic Press: Academic Press, New York
[21] Ramani, A., Inverse scattering, ordinary differential equations of Painlevé type and Hirota’s bilinear formalism, (1980) · Zbl 0599.47017
[22] Sun, Yepeeng; Tam, Honwah, Grammian solutions and pfaffianization of a non-isospectral and variable-coefficient Kadomtsev-Petviashvilli equation, J.Math.Anal.Appl., 343, 810-817, (2008) · Zbl 1147.35088
[23] Sawada, K.; Kotera, T., Prog.Theo.Phys, 51, 1355, (1974)
[24] Shan, Wen-Rui; Yan, Tian-Zhong; Xing, L.; Min, Li; Tian, Bo, Analytic study on the Sawada-Kotera equation with a nonvanishing boundary condition in fluids, Commun Nonlinear Sci Numer Simulat, 18, 1568-1575, (2013) · Zbl 1311.35264
[25] Sato, M.; Noumi, M., Soliton Equations and the Universal Grassmann Manifold, Sugaku Kokyuroku, Sophia Univ, 18, (1984) · Zbl 0541.58001
[26] Ya-Ning, Tang; Wen-Xiu, Ma; Wei, Xu, Grammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation, Chin.Phys.B, 21, 070212, (2012)
[27] Tang, Yaning, Applied Mathematical Modelling, (2013)
[28] Yan, Zhenya, Multiple solution profiles to the higher-dimensional Kadomtsev-Petviashvilli equations via Wronskian determinant, Chaos, Solitons and Fractals., 33, 951-957, (2007) · Zbl 1151.35421
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