Abstract
We compute N-soliton solutions and analyze the Hirota N-soliton conditions, in (2+1)-dimensions, based on the Hirota bilinear formulation. An algorithm to check the Hirota conditions is proposed by comparing degrees of the polynomials generated from the Hirota function in N wave vectors. A weight number is introduced while transforming the Hirota function to achieve homogeneity of the resulting polynomial. Applications to three integrable equations: the (2+1)-dimensional KdV equation, the Kadomtsev–Petviashvili equation, the (2+1)-dimensional Hirota–Satsuma–Ito equation, are made, thereby providing proofs of the existence of N-soliton solutions in the three model equations.
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References
Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Biondini, G., Kodama, Y.: On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy. J. Phys. A Math. Gen. 36, 10519–10536 (2003)
Boiti, M., Leon, J., Manna, M., Pempinelli, F.: On the spectral transform of Korteweg–de Vries equation in two spatial dimensions. Inverse Probl. 2, 271–279 (1986)
Deconinck, B.: Canonical variables for multiphase solutions of the KP equation. Stud. Appl. Math. 104, 229–292 (2000)
Hasegawa, A.: Optical Solitons in Fibers. Springer-Verlag: Berlin Heidelberg and AT & T Bell Laboratories (1989 and 1990)
Hietarinta, J.: A search for bilinear equations passing Hirota’s three-soliton condition. I. KdV-type bilinear equations. J. Math. Phys. 28, 1732–1742 (1987)
Hietarinta, J.: Introduction to the Hirota bilinear method. In: Kosmann-Schwarzbach, Y., Grammaticos, B., Tamizhmani, K.M. (eds.) Integrability of Nonlinear Systems, pp. 95–103. Springer, Berlin (1997)
Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)
Hirota, R.: A new form of Bäcklund transformations and its relation to the inverse scattering problem. Prog. Theor. Phys. 52, 1498–1512 (1974)
Hirota, R.: Direct methods in soliton theory. In: Bullough, R.K., Caudrey, P. (eds.) Solitons, pp. 157–176. Springer, Berlin, Heidelberg (1980)
Hirota, R.: Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)
Hirota, R., Satsuma, J.: \(N\)-soliton solutions of model equations for shallow water waves. J. Phys. Soc. Jpn. 40, 611–612 (1976)
Hosseini, K., Ma, W.X., Ansari, R., Mirzazadeh, M., Pouyanmehr, R., Samadani, F.: Evolutionary behavior of rational wave solutions to the (4+1)-dimensional Boiti—Leon–Manna–Pempinelli equation. Phys. Scr. 95, 065208 (2020)
Inc, M., Hosseini, K., Samavat, M., Mirzazadeh, M., Eslami, M., Moradi, M., Baleanu, D.: \(N\)-wave and other solutions to the B-type Kadomtsev–Petviashvili equation. Therm. Sci. 23(Suppl. 6), S2027–S2035 (2019)
Ito, M.: An extension of nonlinear evolution equations of the K-dV (mK-dV) type to higher orders. J. Phys. Soc. Jpn. 49, 771–778 (1980)
Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)
Liu, W., Wazwaz, A.-M., Zheng, X.X.: High-order breathers, lumps, and semi-rational solutions to the (2+1)-dimensional Hirota–Satsuma–Ito equation. Phys. Scr. 94, 075203 (2019)
Ma, W.X.: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2, 140–144 (2011)
Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys Conf. Ser. 411, 012021 (2013)
Ma, W.X.: Bilinear equations and resonant solutions characterized by Bell polynomials. Rep. Math. Phys. 72, 41–56 (2013b)
Ma, W.X.: Trilinear equations, Bell polynomials, and resonant solutions. Front. Math. China 8, 1139–1156 (2013c)
Ma, W.X., Fan, E.G.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61, 950–959 (2011)
Ma, W.X., Zhang, Y., Tang, Y.N., Tu, J.Y.: Hirota bilinear equations with linear subspaces of solutions. Appl. Math. Comput. 218, 7174–7183 (2012)
Newell, A.C., Zeng, Y.B.: The Hirota conditions. J. Math. Phys. 27, 2016–2021 (1986)
Nizhnik, L.: Integration of multidimensional nonlinear equations by the inverse problem method. Sov. Phys. Dolk. 25, 706–708 (1981)
Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons: The Inverse Scattering Method. Consultants Bureau, New York (1984)
Satsuma, J.: \(N\)-soliton solution of the two-dimensional Kortweg–de Vries equation. J. Phys. Soc. Jpn. 40, 286–290 (1976)
Sawada, K., Kotera, T.: A method for finding \(N\)-soliton solutions of the K.d.V. equation and K.d.V.-like equation. Prog. Theor. Phys. 51, 1355–1367 (1974)
Veselov, A.P., Novikov, S.P.: Finite-zone, two-dimensional, potential Schrödinger operators. Explicit formulas and evolution equations. Sov. Math. Dokl. 30, 588–591 (1984)
Yang, X.Y., Zhang, Z., Li, W.T., Li, B.: Breathers, lumps and hybrid solutions of the (2+1)-dimensional Hirota–Satsuma–Ito equation. Rocky Mountain J. Math. 50, 319–335 (2020)
Acknowledgements
The work was supported in part by NSFC under the Grants 11975145 and 11972291. The author would also like to thank Ahmed Ahmed, Alle Adjiri, Yushan Bai, Liyuan Ding, Yehui Huang, Xing Lü, Solomon Manukure, Morgan McAnally, Fudong Wang and Yong Zhang for their valuable discussions
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Ma, WX. N-soliton solutions and the Hirota conditions in (2+1)-dimensions. Opt Quant Electron 52, 511 (2020). https://doi.org/10.1007/s11082-020-02628-7
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DOI: https://doi.org/10.1007/s11082-020-02628-7