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Characteristics of Abundant Lumps and Interaction Solutions in the (4+1)-Dimensional Nonlinear Partial Differential Equation

  • Xiu-Bin Wang ORCID logo EMAIL logo and Bo Han EMAIL logo
Published/Copyright: January 25, 2020

Abstract

In this work, the (4+1)-dimensional Fokas equation, which is an important physics model, is under investigation. Based on the obtained soliton solutions, the new rational solutions are successfully constructed. Moreover, based on its bilinear formalism, a concise method is employed to explicitly construct its rogue-wave solution and interaction solution with an ansätz function. Finally, the main characteristics of these solutions are graphically discussed. Our results can be helpful for explaining some related nonlinear phenomena.

Acknowledgements

We express our sincere thanks to the editor and reviewer for their valuable comments. This work is supported by the National Key Research and Development Program of China under Grant No. 2017YFB0202901 and the National Natural Science Foundation of China under Grant No. 11871180.

References

[1] G. W. Bluman and S. Kumei, Symmetries and differential equations, in: Grad. Texts in Math, Vol. 81, Springer-Verlag, New York, Heidelberg, Berlin, 1989.10.1007/978-1-4757-4307-4Search in Google Scholar

[2] R. Hirota, Direct Methods in Soliton Theory, Springer, Berlin, 2004.10.1017/CBO9780511543043Search in Google Scholar

[3] M. J. Ablowitz and P. A. Clarkson, Solitons; Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.10.1017/CBO9780511623998Search in Google Scholar

[4] W. Q. Peng, S. F. Tian, X. B. Wang and T. T. Zhang, Riemann–Hilbert method and multi-soliton solutions for three-component coupled nonlinear Schrödinger equations, J. Geom. Phys. 146 (2019), 103508.10.1016/j.geomphys.2019.103508Search in Google Scholar

[5] V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons, Springer, Berlin, 1991.10.1007/978-3-662-00922-2Search in Google Scholar

[6] S. F. Tian, Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation, Appl. Math. Lett. 100 (2020), 106056.10.1016/j.aml.2019.106056Search in Google Scholar

[7] S. F. Tian, Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method, J. Differ. Equ. 262 (2017), 506–558.10.1016/j.jde.2016.09.033Search in Google Scholar

[8] A. M. Wazwaz, (2+1)-dimensional Burgers equations BE(m+n+1): using the recursion operator, Appl. Math. Comput. 219 (2013), 9057–9068.10.1016/j.amc.2013.03.093Search in Google Scholar

[9] E. Inan and D. Kaya, Some Exact Solutions to the Potential Kadomtsev-Petviashvili Equation and to a System of Shallow Water Equation, Phys. Lett. A 35 (2006), 314–322.10.1016/j.physleta.2006.01.106Search in Google Scholar

[10] H. Chen and H. Zhang, New multiple soliton solutions to the general burgers-fisher equation and the Kuramot-Sivashinsky equation, Chaos, Solitons and Fractals 28 (2004), 71–76.10.1016/S0960-0779(03)00081-XSearch in Google Scholar

[11] W. X. Ma and E. G. Fan, Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl. 61 (2011), 950–959.10.1016/j.camwa.2010.12.043Search in Google Scholar

[12] W. X. Ma, R. G. Zhou and L. Gao, Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2+1) dimensions, Mod. Phys. Lett. A 24 (2009), 1677–1688.10.1142/S0217732309030096Search in Google Scholar

[13] W. X. Ma and E. G. Fan, Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl. 61 (2011), 950–959.10.1016/j.camwa.2010.12.043Search in Google Scholar

[14] W. Tan and Z. D. Dai, Dynamics of kinky wave for (3+1)-dimensional potential Yu–Toda–Sasa–Fukuyama equation, Nonlinear Dyn. 85 (2016), 817–823.10.1007/s11071-016-2725-1Search in Google Scholar

[15] P. Muller, C. Garrett and A. Osborne, Rogue waves, Oceanography 18 (2005), 66–75.10.5670/oceanog.2005.30Search in Google Scholar

[16] C. Kharif, E. Pelinovsky and A. Slunyaey, Rogue Waves in the Ocean, observation, theories and modeling, Springer, New York, 2009.Search in Google Scholar

[17] N. Akhmediev, A. Ankiewicz and J. M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schrödinger equation, Phys. Rev. E 80 (2009), 026601.10.1103/PhysRevE.80.026601Search in Google Scholar PubMed

[18] D. R. Solli, C. Ropers, P. Koonath and B. Jalali, Optical rogue waves, Nature 450 (2007), 1054–1057.10.1038/nature06402Search in Google Scholar PubMed

[19] V. Yu. Bludov, V. V. Konotop and N. Akhmediev, Rogue waves as spatial energy concentrators in arrays of nonlinear waveguides, Opt. Lett. 34 (2009), 3015–3017.10.1364/OL.34.003015Search in Google Scholar PubMed

[20] A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin and P. V. E. McClintock, Statistical properties of strongly nonlinear waves within a resonator, Phys. Rev. Lett. 101 (2008), 065303.10.1103/PhysRevLett.101.065303Search in Google Scholar PubMed

[21] A. Montina, U. Bortolozzo, S. Residori and F. T. Arecchi, Rogue waves and their generating mechanisms in different physical contexts, Phys. Rep. 528 (2013), 47–89.10.1016/j.physrep.2013.03.001Search in Google Scholar

[22] Z. Y. Yan, Vector financial rogue waves, Phys. Lett. A 375 (2011), 4274–4279.10.1016/j.physleta.2011.09.026Search in Google Scholar

[23] N. Akhmediev, A. Ankiewicz and M. Taki, Waves that appear from nowhere and disappear without a trace, Phys. Lett. A 373 (2009) 675–678.10.1016/j.physleta.2008.12.036Search in Google Scholar

[24] D. H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Aust. Math. Soc. Ser. B 25 (1983), 16–43.10.1017/S0334270000003891Search in Google Scholar

[25] B. L. Guo, L. M. Ling and Q. P. Liu, Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E 85 (2012), 026607.10.1103/PhysRevE.85.026607Search in Google Scholar PubMed

[26] W. X. Ma and Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differential Equations 264 (2018), 2633–2659.10.1016/j.jde.2017.10.033Search in Google Scholar

[27] S. T. Chen and W. X. Ma, Lump solutions to a generalized Bogoyavlensky–Konopelchenko equation, Front. Math. China, 13 (2018), 525–534.10.1007/s11464-018-0694-zSearch in Google Scholar

[28] M. J. Ablowitz and J. Villarroel, Solutions to the time dependent Schrödinger and the Kadomtsev–Petviashvili equations, Phys. Rev. Lett. 78 (1997), 570.10.1103/PhysRevLett.78.570Search in Google Scholar

[29] U. Bandelow and N. Akhmediev, Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa–Satsuma case, Phys. Lett. A 376 (2012), 1558–1561.10.1016/j.physleta.2012.03.032Search in Google Scholar

[30] L. C. Zhao and J. Liu, Rogue-wave solutions of a three-component coupled nonlinear Schrödinger equation, Phys. Rev. E 87 (2013), 013201.10.1103/PhysRevE.87.013201Search in Google Scholar PubMed

[31] W. P. Zhong, Rogue wave solutions of the generalized one-dimensional Gross-Pitaevskii equation, J. Nonlinear Opt. Phys. Mat. 21 (2012), 1250026.10.1142/S0218863512500269Search in Google Scholar

[32] P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves, J. Phys. A 44 (2011), 435204.10.1088/1751-8113/44/43/435204Search in Google Scholar

[33] C. Y. Qin, S. F. Tian, L. Zou and W. X. Ma, Solitary wave and quasi-periodic wave solutions to a (3+ 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation, Adv. Appl. Math. Mech. 10 (2018), 948–977.10.4208/aamm.OA-2017-0220Search in Google Scholar

[34] X. W. Yan, S. F. Tian, M. J. Dong and L. Zhou, Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+ 1)-dimensional generalized breaking soliton equation, Comput. & Math. Appl. 76 (2018), 179–186.10.1016/j.camwa.2018.04.013Search in Google Scholar

[35] W. X. Ma, Lump-type solutions to the (3+1)-dimensional Jimbo–Miwa equation, Int. J. Nonlin. Sci. Num. 17 (2016), 7–8.10.1515/ijnsns-2015-0050Search in Google Scholar

[36] S. F. Tian, Y. F. Zhang, B. L. Feng and H. Q. Zhang, On the Lie algebras, generalized symmetries and Darboux transformations of the fifth-order evolution equations in shallow water, Chin. Ann. Math. B 36 (2015), 543–560.10.1007/s11401-015-0908-6Search in Google Scholar

[37] X. B. Wang, S. F. Tian, C. Y. Qin and T. T. Zhang, Characteristics of the solitary waves and rogue waves with interaction phenomena in a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation, Appl. Math. Lett. 72 (2017), 58–64.10.1016/j.aml.2017.04.009Search in Google Scholar

[38] X. B. Wang, S. F. Tian, C. Y. Qin and T. T. Zhang, Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation, Appl. Math. Lett. 68 (2017), 40–47.10.1016/j.aml.2016.12.009Search in Google Scholar

[39] X. B. Wang, S. F. Tian, C. Y. Qin and T. T. Zhang, Characteristics of the breathers, rogue waves and solitary waves in a generalized (2+1)-dimensional Boussinesq equation, EPL 115 (2016), 10002.10.1209/0295-5075/115/10002Search in Google Scholar

[40] X. B. Wang and B. Han, The three-component coupled nonlinear Schrödinger equation: Rogue waves on a multi-soliton background and dynamics, EPL 126 (2019), 15001.10.1209/0295-5075/126/15001Search in Google Scholar

[41] K. A. Gepreel, Modified simple equation method to the nonlinear Hirota Satsuma KdV system, J. Inf. Comput. Sci. 10 (2015), 054–062.Search in Google Scholar

[42] D. S. Wang and Y. B. Yin, Symmetry analysis and reductions of the two-dimensional generalized Benney system via geometric approach, Comput. Math. Appl. 71 (2016), 748–757.10.1016/j.camwa.2015.12.035Search in Google Scholar

[43] C. Q. Dai and Y. Y. Wang, Spatiotemporal localizations in (3+1)-dimensional PT-symmetric and strongly nonlocal nonlinear media, Nonlinear Dynam. 83 (2016), 2453–2459.10.1007/s11071-015-2493-3Search in Google Scholar

[44] A. M. Wazwaz, The Hirota’s direct method and the tanh-coth method for multiple-soliton solutions of the Sawada–Kotera–Ito seventh-order equation, Appl. Math. Comput. 199 (2008), 133–138.10.1016/j.amc.2007.09.034Search in Google Scholar

[45] F. J. Yu and Z. Y. Yan, New rogue waves and dark-bright soliton solutions for a coupled nonlinear Schrödinger equation with variable coefficients, Appl. Math. Comput. 233 (2014), 351–358.10.1016/j.amc.2014.02.023Search in Google Scholar

[46] C. Li, J. He and K. Porseizan, Rogue waves of the Hirota and the Maxwell–Bloch equations, Phys. Rev. E 87 (2013), 012913.10.1103/PhysRevE.87.012913Search in Google Scholar PubMed

[47] Z. Xu, H. Chen and Z. Dai, Rogue wave for the (2+1)-dimensional Kadomtsev–Petviashvili equation, Appl. Math. Lett. 37 (2014), 34–38.10.1016/j.aml.2014.05.005Search in Google Scholar

[48] A. S. Fokas, Integrable nonlinear evolution partial differential equations in 4+2 and 3+1 dimensions, Phys. Rev. Lett. 96 (2006), 190201.10.1103/PhysRevLett.96.190201Search in Google Scholar PubMed

[49] S. Sheng, C. Tian and W. Y. Qian, Bilinearization and new multisoliton solutions for the (4+1)-dimensional Fokas equation, Pramana-J, Phys. 86 (2016), 1259–1267.10.1007/s12043-015-1173-7Search in Google Scholar

[50] J. Lee, R. Sakthivel and L. Wazzan, Exact traveling wave solutions of (4+1)-dimensional nonlinear Fokas equation, Mod. Phys. Lett. B 24 (2010), 1011.10.1142/S0217984910023062Search in Google Scholar

[51] Z. Z. Zhang and Z. Y. Yan, Symmetry groups and exact solutions of new (4+1)-dimensional nonlinear Fokas equation, Commun. Theor. Phys. 51 (2009), 876–880.10.1088/0253-6102/51/5/24Search in Google Scholar

[52] W. Tan, Z. D. Dai, J. L. Xie and D. Q. Qiu, Parameter limit method and its application in the (4+1)-dimensional Fokas equation, Comput. Math. Appl. 75 (2018), 4214–4220.10.1016/j.camwa.2018.03.023Search in Google Scholar

[53] L. Cheng and Y. Zhang, Lump-type solutions for the (4+1)-dimensional Fokas equation via symbolic computations, Mod. Phys. Lett. B 31 (2017), 1750224.10.1142/S0217984917502244Search in Google Scholar

[54] Y. H. He, Exact solutions for (4+1)-dimensional nonlinear Fokas equation using extended F-expansion method and its variant, Math. Prob. Engin. 2014 (2014), 972519.10.1155/2014/972519Search in Google Scholar

[55] X. B. Wang, S. F. Tian, L. L. Feng and T. T. Zhang, On quasi-periodic waves and rogue waves to the (4+1)-dimensional nonlinear Fokas equation, J. Math. Phys. 59 (2018), 073505.10.1063/1.5046691Search in Google Scholar

Received: 2018-12-20
Accepted: 2019-12-30
Published Online: 2020-01-25
Published in Print: 2020-05-26

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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