Lump and rogue wave solutions to a (2+1)-dimensional Boussinesq type equation. (English) Zbl 1469.35084
Summary: In this paper, we study a (2+1)-dimensional Boussinesq type equation. By applying the Hirota direct method, lump and line rogue wave solutions are presented with the aid of symbolic computations. The solutions are expressed in terms of a set of restricted parameters with necessary and sufficient conditions that guarantee their existence. An interesting result is that when the parameters meet the rank requirement, we have lump solutions, otherwise, we may get line rogue waves.
MSC:
| 35C11 | Polynomial solutions to PDEs |
| 35C07 | Traveling wave solutions |
| 35C08 | Soliton solutions |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q51 | Soliton equations |
References:
| [1] | Ablowitz, M. J.; Satsuma, J., Solitons and rational solutions of nonlinear evolution equations, J. Math. Phys., 19, 10, 2180-2186 (1978) · Zbl 0418.35022 |
| [2] | Akhmediev, N.; Ankiewicz, A.; Soto-Crespo, J. M., Rogue waves and rational solutions of the nonlinear Schrödinger equation, Phys. Rev. E, 80, Article 026601 pp. (2009) |
| [3] | Boussinesq, J., Théorie de l’intumescence liquide, appelée onde solitaire ou de translation se propagente dans un canal rectangulaire, Comptes Rendus, 72, 755-759 (1871) · JFM 03.0486.01 |
| [4] | Boussinesq, J., Théorie de ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblemant parielles de la surface au fond, J. Pure Appl., 17, 55-108 (1872) · JFM 04.0493.04 |
| [5] | Cheng, L.; Zhang, Y., Wronskian and linear superposition solutions to generalized KP and BKP equations, Nonlinear Dyn., 90, 1, 355-362 (2017) · Zbl 1390.37111 |
| [6] | Clarkson, P. A.; Dowie, E., Rational solutions of the Boussinesq equation and applications to rogue waves, Trans. Math. Its Appl., 1, 1 (2017), tnx003 · Zbl 1403.37073 |
| [7] | Drazin, P. G.; Johnson, R. S., Solitons: An Introduction (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0661.35001 |
| [8] | Dubard, P.; Matveev, V. B., Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation, Nat. Hazards Earth Syst. Sci., 11, 667-672 (2011) |
| [9] | Dubard, P.; Gaillard, P.; Klein, C.; Matveev, V. B., Multi-rogue wave solutions of the NLS equation and positon solutions of the KdV equation, Eur. Phys. J. Spec. Top., 185, 247-258 (2010) |
| [10] | Fokas, A. S.; Ablowitz, M. J., On the inverse scattering transform of multidimensional nonlinear equations related to first-order systems in the plane, J. Math. Phys., 25, 8, 2494-2505 (1984) · Zbl 0557.35110 |
| [11] | Gao, L. N.; Zi, Y. Y.; Yin, Y. H.; Ma, W. X.; Lü, X., Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation, Nonlinear Dyn., 89, 3, 2233-2240 (2017) |
| [12] | Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27, 1192-1194 (1971) · Zbl 1168.35423 |
| [13] | Hirota, R., Exact solution of the modified Korteweg-de Vries equation for multiple collisions of solitons, J. Phys. Soc. Jpn., 33, 1456-1458 (1972) |
| [14] | Hirota, R., Exact solution of the sine-Gordon equation for multiple collisions of solitons, J. Phys. Soc. Jpn., 33, 1459-1463 (1972) |
| [15] | Hirota, R., The Direct Method in Soliton Theory (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1099.35111 |
| [16] | Kaup, D. J., The lump solutions and the Bäcklund transformation for the three-dimensional three-wave resonant interaction, J. Math. Phys., 22, 6, 1176-1181 (1981) · Zbl 0467.35070 |
| [17] | Kaur, L.; Wazwaz, A. M., Lump, breather and solitary wave solutions to new reduced form of the generalized BKP equation, Int. J. Numer. Methods Heat Fluid Flow, 29, 2, 569-579 (2019) |
| [18] | Liang, Y.; Wei, G.; Li, X., Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation, Nonlinear Dyn., 62, 1-2, 195-202 (2010) · Zbl 1208.35018 |
| [19] | Lü, X., New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada-Kotera model, Nonlinear Dyn., 76, 161-168 (2014) · Zbl 1319.35222 |
| [20] | Lü, X.; Ma, W. X., Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation, Nonlinear Dyn., 85, 1217-1222 (2016) · Zbl 1355.35159 |
| [21] | Lü, X.; Wang, J. P.; Lin, F. H.; Zhou, X. W., Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water, Nonlinear Dyn., 91, 1249-1259 (2018) |
| [22] | Ma, H. C.; Deng, A. P., Lump solution of (2+1)-dimensional Boussinesq equation, Commun. Theor. Phys., 65, 546-552 (2016) · Zbl 1338.35371 |
| [23] | Ma, W. X., Wronskians, generalized Wronskians and solutions to the Korteweg-de Vries equation, Chaos Solitons Fractals, 19, 163-170 (2004) · Zbl 1068.35138 |
| [24] | Ma, W. X., Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379, 1975-1978 (2015) · Zbl 1364.35337 |
| [25] | Ma, W. X., N-soliton solutions and the Hirota conditions in (2+1)-dimensions, Opt. Quantum Electron., 52, 511 (2020) |
| [26] | Ma, W. X., N-soliton solutions and the Hirota conditions in (1+1)-dimensions, Int. J. Nonlinear Sci. Numer. Simul., 22 (2021), accepted for publication |
| [27] | Ma, W. X., N-soliton solution of a combined pKP-BKP equation, J. Geom. Phys., 165, Article 104191 pp. (2021) · Zbl 1470.35310 |
| [28] | Ma, W. X.; Zhou, Y., Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differ. Equ., 264, 2633-2659 (2018) · Zbl 1387.35532 |
| [29] | Ma, W. X.; Li, C. X.; He, J., A second Wronskian formulation of the Boussinesq equation, Nonlinear Anal., Theory Methods Appl., 70, 12, 4245-4258 (2009) · Zbl 1159.37425 |
| [30] | Ma, W. X.; Zhou, Y.; Dougherty, R., Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations, Int. J. Mod. Phys. B, 30, Article 1640018 pp. (2016) · Zbl 1375.37162 |
| [31] | Ma, W. X.; Zhang, Y.; Tang, Y., Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms, East Asian J. Appl. Math., 10, 4, 732-745 (2020) · Zbl 1464.35289 |
| [32] | Manakov, S. V.; Zakhorov, V. E.; Bordag, L. A.; Its, A. R.; Matveev, V. B., Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. A, 63, 205-206 (1977) |
| [33] | Manukure, S.; Zhou, Y., A (2+1)-dimensional shallow water equation and its explicit lump solutions, Int. J. Mod. Phys. B, 33, 7, Article 1950038 pp. (2019) · Zbl 1425.35159 |
| [34] | Manukure, S.; Zhou, Y.; Ma, W. X., Lump solutions to a (2+1)-dimensional extended KP equation, Comput. Math. Appl., 75, 7, 2414-2419 (2018) · Zbl 1409.35183 |
| [35] | Manukure, S.; Chowdhury, A.; Zhou, Y., Complexiton solutions to the asymmetric Nizhnik-Novikov-Veselov equation, Int. J. Mod. Phys. B, 33, Article 1950098 pp. (2019) · Zbl 1427.35237 |
| [36] | Matsuno, Y., Exact multi-soliton solution of the Benjamin-Ono equation, J. Phys. A, Math. Gen., 12, 4, 619-621 (1979) · Zbl 0399.45014 |
| [37] | Matveev, V.; Salle, M., Darboux Transformations and Solitons (1991), Springer: Springer Berlin · Zbl 0744.35045 |
| [38] | McAnally, M.; Ma, W. X., Explicit solutions and Darboux transformations of a generalized D-Kaup-Newell hierarchy, Nonlinear Dyn., 102, 2767-2782 (2020) · Zbl 1517.37068 |
| [39] | Peregrine, D., Water waves, nonlinear Schrödinger equations and their solutions, J. Aust. Math. Soc. Ser. B, 25, 1, 16-43 (1983) · Zbl 0526.76018 |
| [40] | Satsuma, J.; Ablowitz, M. J., Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 20, 7, 1496-1503 (1979) · Zbl 0422.35014 |
| [41] | Satsuma, J.; Ishimori, Y., Periodic wave and rational soliton solutions of the Benjamin-Ono equation, J. Phys. Soc. Jpn., 46, 2, 681-687 (1979) |
| [42] | Solli, D. R.; Ropers, C.; Koonath, P.; Jalali, B., Optical rogue waves, Nature, 450, 1054-1057 (2007) |
| [43] | Sun, B.; Wazwaz, A. M., Interaction of lumps and dark solitons in the Mel’nikov equation, Nonlinear Dyn., 92, 2, 2049-2059 (2018) · Zbl 1398.37067 |
| [44] | Tajiri, M.; Murakami, Y., Rational growing mode: exact solutions to the Boussinesq equation, J. Phys. Soc. Jpn., 60, 8, 2791-2792 (1991) |
| [45] | Wang, H.; Wang, Y.; Ma, W. X.; Temuer, C., Lump solutions of a new extended (2+1)-dimensional Boussinesq equation, Mod. Phys. Lett. B, 32, Article 1850376 pp. (2018) |
| [46] | Wazwaz, A. M., Painlevé analysis for a new integrable equation combining the modified Calogero—Bogoyavlenskii—Schiff (MCBS) equation with its negative-order form, Nonlinear Dyn., 92, 877-883 (2018) · Zbl 1390.37117 |
| [47] | Yang, J. Y.; Ma, W. X.; Khalique, C., Determining lump solutions for a combined soliton equation in (2+1)-dimensions, Eur. Phys. J. Plus, 135, 494 (2020) |
| [48] | Zhou, Y.; Ma, W. X., Applications of linear superposition principle to resonant solitons and complexitons, Comput. Math. Appl., 73, 1697-1706 (2017) · Zbl 1372.35069 |
| [49] | Zhou, Y.; Ma, W. X., Complexiton solutions to nonlinear partial differential equations by the direct method, J. Math. Phys., 58, Article 101511 pp. (2017) · Zbl 1382.37073 |
| [50] | Zhou, Y.; Manukure, S., Complexiton solutions to the Hirota-Satsuma-Ito equation, Math. Methods Appl. Sci., 42, 1-8 (2019) · Zbl 1411.37060 |
| [51] | Zhou, Y.; Manukure, S.; Ma, W. X., Lump and lump-soliton solutions to the Hirota-Satsuma-Ito equation, Commun. Nonlinear Sci. Numer. Simul., 68, 56-62 (2019) · Zbl 1509.35101 |
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