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Lump solutions of a nonlinear PDE containing a third-order derivative of time. (English) Zbl 07281327
Summary: A nonlinear partial differential equation combining with a third-order derivative of the time variable \(D_x D_t^3\) is studied. By adding a new fourth-order derivative term, its lump solutions are explicitly constructed by the Hirota bilinear method and symbolic computation. Furthermore, the effect of the new fourth-order derivative term on the solution is discussed. The dynamical behaviors of two particular lump solutions are analyzed with different choices of the parameters.
65 Numerical analysis
34 Ordinary differential equations
Full Text: DOI
[1] Ablowitz, M. J.; Segur, H., Solitons and the Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia · Zbl 0472.35002
[2] Hirota, R., The Direct Method in Soliton Theory (2004), Cambridge University Press
[3] Novikov, S.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of Solitons the Inverse Scattering Method (1984), Consultants Bureau: Consultants Bureau New York · Zbl 0598.35002
[4] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[5] Ma, W. X.; Zhou, Y., Lump solutions to nonlinear differential equations via hirota bilinear forms, J. Differential Equations, 264, 2633-2659 (2018) · Zbl 1387.35532
[6] Ma, W. X.; Zhou, Y.; Dougherty, R., Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations, Internat. J. Modern Phys. B, 30, Article 1640018 pp. (2016) · Zbl 1375.37162
[7] Tan, W.; Dai, H. P.; Dai, Z. D.; Zhong, W. Y., Emergence and space-time structure of lump solution to the (2+1)-dimensional generalized KP equation, Pramana-J. Phys., 77, 89 (2017)
[8] Satsuma, J.; Ablowitz, M. J., Two-dimensional lumps in nonlinear dispersive systems, J. Math. Phys., 20, 1496-1503 (1979) · Zbl 0422.35014
[9] Caudrey, P. J., Memories of Hirota’s method: application to the reduced Maxwell-Bloch system in the early 1970s, Phil. Trans. R. Soc. A, 369, 1215-1227 (2011) · Zbl 1219.37046
[10] Hietarinta, J., Introduction to the Hirota bilinear method, (Kosmann-Schwarzbach, Y.; Grammaticos, B.; Tamizhmani, K. M., Integrability of Nonlinear Systems (1997), Springer: Springer Berlin), 95-103 · Zbl 0907.58030
[11] Ma, W. X., Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379, 1975-1978 (2015) · Zbl 1364.35337
[12] Manakov, S. V.; Zakharov, V. E.; Bordag, L. A.; Matveev, V. B., Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett. A, 63, 205-206 (1977)
[13] Kaup, D. J., The lump solutions and the Bäcklund transformation for the three-dimensional three-wave resonant interaction, J. Math. Phys., 22, 1176-1181 (1981) · Zbl 0467.35070
[14] Imai, K., Dromion and lump solutions of the Ishimori-I equation, Progr. Theoret. Phys., 98, 1013-1023 (1997)
[15] Gilson, C. R.; Nimmo, J. J.C., Lump solutions of the BKP equation, Phys. Lett. A, 147, 472-476 (1990)
[16] Yang, J. Y.; Ma, W. X., Lump solutions of the BKP equation by symbolic computation, Int. J. Mod. Phys. B, 30, Article 1640028 pp. (2016) · Zbl 1357.35080
[17] Yong, X. L.; Ma, W. X.; Huang, Y. H.; Liu, Y., Lump solutions to the Kadomtsev-Petviashvili I equation with a self-consistent source, Comput. Math. Appl., 75, 3414-3419 (2018) · Zbl 1409.35187
[18] Chen, S. T.; Ma, W. X., Lumps solutions to a generalized Calogero-Bogoyavlenskii-Schiff equation, Comput. Math. Appl., 76, 1680-1685 (2018) · Zbl 1434.35153
[19] Chen, S. T.; Ma, W. X., Lump solutions to a generalized Bogoyavlensky-Konopelchenko equation, Front. Math. China, 13, 525-534 (2018) · Zbl 1403.35259
[20] Lü, X.; Chen, S. T.; Ma, W. X., Constructing lump solutions to a generalized Kadomtsev-Petviashvili-Boussinesq equation, Nonlinear Dynam., 86, 523-534 (2016) · Zbl 1349.35007
[21] Ma, W. X., A search for lump solutions to a combined fourth-order nonlinear PDE in (2+1)-dimensions, J. Appl. Anal. Comput., 9, 1319-1332 (2019)
[22] Ma, W. X.; Qin, Z. Y.; Lü, X., Lump solutions to dimensionally reduced p-gKP and p-gBKP equations, Nonlinear Dynam., 84, 923-931 (2016) · Zbl 1354.35127
[23] Ma, W. X.; Yong, X. L.; Zhang, H. Q., Diversity of interaction solutions to the (2+1)- dimensional Ito equation, Comput. Math. Appl., 75, 289-295 (2018) · Zbl 1416.35232
[24] Zhang, H. Q.; Ma, W. X., Lump solutions to the (2+1)-dimensional Sawada-Kotera equation, Nonlinear Dynam., 87, 2305-2310 (2017)
[25] Chen, Q. X.; Ma, W. X.; Huang, Y. H., Study of lump solutions to an extended Calogero-Bogoyavlenskii-Schiff equation involving three fourth-order terms, Phys. Scr., 95, Article 095207 pp. (2020)
[26] Yang, J. Y.; Ma, W. X.; Qin, Z. Y., Lump and lump-soliton to the (2+1)-dimensional Ito equation, Anal. Math. Phys., 8, 427-436 (2018) · Zbl 1403.35261
[27] Tang, Y. N.; Tao, S. Q.; Qing, G., Lump solutions and interaction phenomenon of them for two classes of nonlinear evolution equations, Comput. Math. Appl., 72, 2334-2342 (2016) · Zbl 1372.35268
[28] Zhang, J. B.; Ma, W. X., Mixed lump-kink solutions to the BKP equation, Comput. Math. Appl., 74, 591-596 (2017) · Zbl 1387.35540
[29] Kofane, T. C.; Fokou, M.; Mohamadou, A.; Yomba, E., Lump solutions and interaction phenomenon to the third-order nonlinear evolution equation, Eur. Phys. J. Plus, 132, 465-473 (2017)
[30] Zhao, H. Q.; Ma, W. X., Mixed lump-kink solutions to the KP equation, Comput. Math. Appl., 74, 1399-1405 (2017) · Zbl 1394.35461
[31] Ma, W. X., Lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation, Int. J. Nonlinear Sci. Number. Simul., 17, 355-359 (2017) · Zbl 1401.35273
[32] Yang, J. Y.; Ma, W. X., Abundant lump-type solutions of the Jimbo-Miwa equation in (3+1)-dimensions, Comput. Math. Appl., 73, 220-225 (2017) · Zbl 1368.35238
[33] Qi, F. H.; Ma, W. X.; Qu, Q. X.; Wang, P., Lump-type and interaction solutions to an extended (3+1)-dimensional Jimbo-Miwa equation, Internat. J. Modern Phys. B, 34, Article 2050043 pp. (2020) · Zbl 1434.35169
[34] Ma, W. X., Lump and interaction solutions to linear PDEs in 2+1 dimensions via symbolic computation, Modern Phys. Lett. B, 33, Article 1950457 pp. (2019)
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