A symbolic computation approach to constructing rogue waves with a controllable center in the nonlinear systems. (English) Zbl 1409.76017
Summary: A symbolic computation approach to constructing higher order rogue waves with a controllable center of the nonlinear systems is presented, making use of their Hirota bilinear forms. As some examples, it turns out that some higher order rogue wave solutions of the Kadomtsev-Petviashvili (KP) type equations in \((3+1)\) and \((2+1)\)-dimensions are obtained. Some features of controllable center of rogue waves are graphically discussed.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76B25 | Solitary waves for incompressible inviscid fluids |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
References:
| [1] | Osborne, A. R., Nonlinear Ocean Waves (2009), Academic Press: Academic Press New York |
| [2] | Akhmediev, N.; Eleonskii, V. M.; Kulagin, N. E., Exact first-order solutions of the nonlinear Schrödinger equation, Theoret. Math. Phys., 72, 809-818 (1987) · Zbl 0656.35135 |
| [3] | Akhmediev, N.; Ankiewicz, A.; Taki, M., Waves that appear from nowhere and disappear without a trace, Phys. Lett. A, 373, 675-678 (2009) · Zbl 1227.76010 |
| [4] | Müller, P.; Garrett, C.; Osborne, A., Rogue waves, Oceanography, 18, 66-75 (2005) |
| [5] | Kharif, C.; Pelinovsky, E.; Slunyaev, A., (Rogue Waves in the Ocean. Rogue Waves in the Ocean, Advances in Goephysical and Enviromental Mechnics and Mathematics (2009), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1230.86001 |
| [6] | Solli, D. R.; Ropers, C.; Koonath, P.; Jalali, B., Optical rogue waves, Nature, 450, 1045-1057 (2007) |
| [7] | Kibler, B.; Fatome, J.; Finot, C., The Peregrine soliton in nonlinear fibre optics, Nat. Phys., 6, 10, 790-795 (2010) |
| [8] | Chabchoub, A.; Hoffmann, N. P.; Akhmediev, N., Rogue wave observation in a water wave tank, Phys. Rev. Lett., 106, 20, 204502 (2011) |
| [9] | Vinayagam, P. S.; Radha, R.; Porsezian, K., Taming rogue waves in Vector BECs, Phys. Rev. A, 88, 042906 (2013) |
| [10] | Yan, Z. Y., Financial rogue waves, Commun. Theor. Phys., 54, 5, 947 (2010) · Zbl 1219.91143 |
| [11] | Yan, Z. Y., Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients, Phys. Lett. A, 374, 672-679 (2010) · Zbl 1235.35266 |
| [12] | Zhang, Y.; Nie, X. J.; Zha, Q. L., Rogue wave solutions for the Heisenberg ferromagnet equation, Chin. Phys. Lett., 31, 060201 (2014) |
| [13] | Zhaqilao, Rogue waves and rational solutions of a \((3 + 1)\)-dimensional nonlinear evolution equation, Phys. Lett. A, 377, 3021-3026 (2013) · Zbl 1370.35243 |
| [14] | Xu, Z. H.; Chen, H. L.; Dai, Z. D., Rogue wave for the \((2 + 1)\)-dimensional Kadomtsev-Petviashvili equation, Appl. Math. Lett., 37, 34-38 (2014) · Zbl 1314.35155 |
| [15] | Ma, W. X., Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379, 1975-1978 (2015) · Zbl 1364.35337 |
| [16] | P.A. Clarkson, E. Dowie, Rational solutionas of the Boussinesq equation and applications to rogue waves, 2017. arXiv:1609.00503v2; P.A. Clarkson, E. Dowie, Rational solutionas of the Boussinesq equation and applications to rogue waves, 2017. arXiv:1609.00503v2 · Zbl 1403.37073 |
| [17] | Gaillard, P., Rational solutions to the KPI equation and multi rogue waves, Ann. Phys., 367, 1-5 (2016) · Zbl 1378.35266 |
| [18] | Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press · Zbl 0762.35001 |
| [19] | Zhaqilao; Li, Z. B., New multi-soliton solutions for the \((2 + 1)\)-dimensional Kadomtsev-Petviashvili equation, Commun. Theor. Phys., 49, 585-589 (2008) · Zbl 1392.37075 |
| [20] | Zhaqilao; Li, Z. B., Periodic-soliton solutions of the \((2 + 1)\)-dimensional Kadomtsev-Petviashvili equation, Chin. Phys. B, 17, 2333-2338 (2008) |
| [21] | P. Dubard, V.B. Matveev, Multi-rogue waves solutions: from the NLS to the KP-I equation, 26 (2013) R93-R125.; P. Dubard, V.B. Matveev, Multi-rogue waves solutions: from the NLS to the KP-I equation, 26 (2013) R93-R125. · Zbl 1286.35226 |
| [22] | Hiroat, R., The Direct Method in Soliton Theory (2004), Cambridge University Press · Zbl 1099.35111 |
| [23] | Ma, W. X.; Abdeljabbar, A., A bilinear Bäcklund transformation of a \((3 + 1)\)-dimensional generalized KP equation, Appl. Math. Lett., 25, 1500-1504 (2012) · Zbl 1248.37070 |
| [24] | Wazwaz, A. M., Multiple-soliton solutions for a \((3 + 1)\)-dimensional generalized KP equation, Commun. Nonlinear Sci. Numer. Simul., 17, 491-495 (2012) · Zbl 1245.35104 |
| [25] | Ma, W. X.; Zhou, Y., Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations, 264, 2633-2659 (2018) · Zbl 1387.35532 |
| [26] | Yang, J. Y.; Ma, W. X.; Qin, Z. Y., Lum and Lum-solitins to the \((2 + 1)\)-dimensuinal Ito equation., Anal. Math. Phys., (2017) |
| [27] | Zhang, H. Q.; Ma, W. X., Lump solutions to the \((2 + 1)\)-dimensional Sawada-Kotera equation, Nonlinear Dynam., 87, 2305-2310 (2017) |
| [28] | Wang, X. C.; He, J. S.; Li, Y. S., Rogue wave with a controllable center of nonlinear Schrödinger equation, Commun. Theor. Phys., 56, 631-637 (2011) · Zbl 1247.35155 |
| [29] | Ma, W. X.; Li, C. X.; He, J. S., A second Wronskian formulation of the Boussinesq equation, Nonlinear Anal., 70, 4245-4258 (2009) · Zbl 1159.37425 |
| [30] | Wazwaz, A. M., Two B-type Kadomtsev-Petviashvili equations of \((2 + 1)\) and \((3 + 1)\) dimensions: Multiple soliton solutions, rational solutions and periodic solutions, Comput. & Fluids, 86, 357-362 (2013) · Zbl 1290.35028 |
| [31] | Xu, G. Q., The soliton solutions, dromins of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in \((3 + 1)\)-dimensions, Chaos Solitons Fractals, 30, 71-76 (2006) · Zbl 1141.35444 |
| [32] | Wang, M. L.; Zhang, J. L.; Li, X. Z., Decay mode solutions to cylindrical KP equation, Appl. Math. Lett., 62, 29-34 (2016) · Zbl 1356.35211 |
| [33] | Shi, C. G.; Zhao, B. Z.; Ma, W. X., Exact rational solutions to a Boussinesq-like equation in (1+1)-dimensions, Appl. Math. Lett., 48, 170-176 (2015) · Zbl 1326.35064 |
| [34] | Ma, W. X., Generalized bilinear differential equations, Studies in Nonlinear Sciences, 2, 4, 140-144 (2011) |
| [35] | Zhang, H. Q.; Chen, J., Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation, Modern Phys. Lett. B, 30, 10, 1650106 (2016) |
| [36] | Wen, L. L.; Zhang, H. Q., Rogue wave solutions of the \((2 + 1)\)-dimensional derivative nonlinear Schrödinger equation, Nonlinear Dynam., 86, 877-889 (2016) · Zbl 1349.37070 |
| [37] | Zhang, H. Q.; Liu, X. L.; Wen, L. L., Soliton, breather, and rogue Wave for a \((2 + 1)\)-dimensional nonlinear Schrödinger equation, Z. Naturforsch., 71, (2)a, 95-101 (2016) |
| [38] | Yang, J. Y.; Ma, W. X., Abundant interaction solutions of the KP equation, Nonlinear Dynam., 89, 1539-1544 (2017) |
| [39] | Zhao, H. Q.; Ma, W. X., Mixed lump-kink solutions to the KP equation, Comput. Math. Appl., 74, 1399-1405 (2017) · Zbl 1394.35461 |
| [40] | Zhang, J. B.; Ma, W. X., Mixed lump-kink solutions to the BKP equation, Comput. Math. Appl., 74, 591-596 (2017) · Zbl 1387.35540 |
| [41] | Ma, W. X.; Yong, X. L.; Zhang, H. Q., Diversity of interaction solutions to the \((2 + 1)\)-dimensional Ito equation, Comput. Math. Appl. (2017) |
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