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Stability analysis, soliton waves, rogue waves and interaction phenomena for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, China

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Shou-Fu Tian

http://orcid.org/0000-0003-0888-777X

School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou 221116, China

E-mail Address: shoufu2006@126.com; sftian@cumt.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0217984918503451Cited by:3 (Source: Crossref)

Abstract

In this paper, the (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation is discussed, which can be used to describe certain characteristics of soliton in a nonlinear media with weak dispersion. By using the virtue of Bell polynomial, we construct the exact bilinear formalism and soliton wave of the equation, respectively. We also analyze its stability analysis. Moreover, based on the resulting bilinear formalism, we obtain its rouge wave solutions with a direct method. Finally, we also discuss the interaction phenomena between solitary wave solutions and rogue wave solutions. It is hoped that our results can be used to enrich the dynamics of the (3+1)-dimensional nonlinear wave fields.

Keywords:

References

1. B. B. Kadomtsev and V. L. Petviashvili, Sov. Phys. Dokl. 15 (1970) 539. ADS, Google Scholar
2. V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons (Springer, 1991). Crossref, Google Scholar
3. J. J. C. Nimmo, Darboux Transformations from Reductions of the KP Hierarchy (World Scientific, Singapore, 1995). Google Scholar
4. R. Hirota, Direct Methods in Soliton Theory (Springer, 2004). Crossref, Google Scholar
5. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambrige University Press, 1991). Crossref, Google Scholar
6. Z. N. Zhu, Acta Phys. Sin. 41 (1992) 1561. Crossref, Google Scholar
7. W. X. Ma, T. W. Huang and Y. Zhang, Phys. Scripta 82 (2010) 065003. Crossref, Web of Science, Google Scholar
8. W. X. Ma and Y. You, T. Am. Math. Soc. 357 (2005) 1753. Crossref, Web of Science, Google Scholar
9. Y. Chen, Z. Y. Yan and H. Zhang, Phys. Lett. A 307 (2003) 107. Crossref, Web of Science, ADS, Google Scholar
10. B. Li, Y. Chen and Y. Li, Z. Naturforsch. A 63 (2008) 763; M. Chen, X. Li, Y. Wang and B. Li, Commun. Theor. Phys. 67 (2017) 595; X. Li, Y. Wang, M. D. Chen and B. Li, Adv. Math. Phys. 2017 (2017) 1743789. Google Scholar
11. Y. Wang, M. Chen, X. Li and B. Li, Z. Naturforsch. A 72(5) (2017) 419; Y. Liu and B. Li, Chinese J. Phys. 54 (2016) 718; M. D. Chen, X. Li, Y. Wang and B. Li, Commun. Theor. Phys. 67 (2017) 595. Google Scholar
12. P. Muller and C. Garrett, Oceanography 18 (2005) 66. Crossref, Google Scholar
13. Y. Ohta and J. K. Yang, Proc. R. Soc. A 468 (2012) 1716. Crossref, ADS, Google Scholar
14. C. Kharif, E. Pelinovsky and A. Slunyaey, Rogue Waves in the Ocean, Observation, Theories and Modeling (Springer, New York, 2009). Google Scholar
15. N. Akhmediev, A. Ankiewicz and J. M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601. Crossref, Web of Science, ADS, Google Scholar
16. D. R. Solli, C. Ropers, P. Koonath and B. Jalali, Nature 450 (2007) 1054. Crossref, Web of Science, ADS, Google Scholar
17. A. Montina, U. Bortolozzo, S. Residori and F. T. Arecchi, Phys. Rep. 528 (2013) 47. Crossref, Web of Science, ADS, Google Scholar
18. Z. Y. Yan, Phys. Lett. A 375 (2011) 4274. Crossref, Web of Science, ADS, Google Scholar
19. D. H. Peregrine, J. Aust. Math. Soc. Ser. B 25 (1983) 16. Crossref, Google Scholar
20. B. L. Guo, L. M. Ling and Q. P. Liu, Phys. Rev. E 85 (2012) 026607. Crossref, Web of Science, ADS, Google Scholar
21. M. J. Ablowitz and J. Villarroel, Phys. Rev. Lett. 78 (1997) 570. Crossref, Web of Science, ADS, Google Scholar
22. L. C. Zhao and J. Liu, Phys. Rev. E. 87 (2013) 013201. Crossref, Web of Science, ADS, Google Scholar
23. P. Gaillard, J. Phys. A: Math. Theor. 44 (2011) 435204. Crossref, Web of Science, ADS, Google Scholar
24. X. B. Wang, S. F. Tian, C. Y. Qin and T. T. Zhang, Europhys. Lett. 115 (2016) Article 10002; X. B. Wang, S. F. Tian, H. Yan and T. T. Zhang, Comput. Math. Appl. 74 (2017) 556. Google Scholar
25. X. B. Wang, S. F. Tian, C. Y. Qin and T. T. Zhang, Appl. Math. Lett. 68 (2017) 40. Crossref, Web of Science, Google Scholar
26. J. S. He, S. W. Xu and K. Porseizan, Phys. Rev. E 86 (2012) 066603. Crossref, Web of Science, ADS, Google Scholar
27. L. L. Feng and T. T. Zhang, Appl. Math. Lett. 78 (2018) 133; X. B. Wang, T. T. Zhang and M. J. Dong, ibid. 86 (2018) 298; S. F. Tian, ibid. 83 (2018) 65. Google Scholar
28. C. Y. Qin, S. F. Tian, L. Zou and W. X. Ma, Adv. Appl. Math. Mech. 10(4) (2018) 948; X. W. Yan, S. F. Tian, M. J. Dong, L. Zhou and T. T. Zhang, Comput. Math. Appl. 76(1) (2018) 179; C. Y. Qin, S. F. Tian, X. B. Wang, T. T. Zhang and J. Li, ibid. 75(12) (2018) 4221. Google Scholar
29. X. W. Yan, S. F. Tian, M. J. Dong, X. B. Wang and T. T. Zhang, Z. Naturforsch. A 73(5) (2018) 399; S. F. Tian, Commun. Pure Appl. Anal. 17(3) (2018) 923. Google Scholar
30. M. J. Dong, S. F. Tian, X. W. Yan and L. Zou, Comput. Math. Appl. 75(3) (2018) 957; X. B. Wang, S. F. Tian and T. T. Zhang, Proc. Am. Math. Soc. 146(8) (2018) 3353; S. F. Tian and T. T. Zhang, ibid. 146(4) (2018) 1713. Google Scholar
31. X. B. Wang, S. F. Tian, L. L. Feng and T. T. Zhang, J. Math. Phys. 59(7) (2018) 073505; W. Q. Peng, S. F. Tian and T. T. Zhang, Phys. Lett. A 382(38) (2018) 2701. Google Scholar
32. L. L. Feng, S. F. Tian, X. B. Wang and T. T. Zhang, Appl. Math. Lett. 65 (2017) 90; M. J. Xu, S. F. Tian, J. M. Tu, P. L. Ma and T. T. Zhang, Nonlinear Dynam. 82 (2015) 2031. Google Scholar
33. X. B. Wang, S. F. Tian, C. Y. Qin and T. T. Zhang, Appl. Math. Lett. 72 (2017) 58; J. M. Tu, S. F. Tian, M. J. Xu, X. Q. Song and T. T. Zhang, Nonlinear Dynam. 83 (2016) 1199. Google Scholar
34. Z. Xu, H. Chen and Z. Dai, Appl. Math. Lett. 37 (2014) 34. Crossref, Web of Science, Google Scholar
35. Y. Chen, Z. Y. Yan and H. Zhang, Phys. Lett. A 307 (2003) 107; Y. Wang and Y. Chen, J. Math. Anal. Appl. 400 (2013) 624. Google Scholar
36. J. C. Chen, Y. Chen, B. F. Feng and K.-I. Maruno, Phys. Lett. A 379 (2015) 1510. Crossref, Web of Science, ADS, Google Scholar
37. S. F. Tian, J. Phys. A, Math. Theor. 50 (2017) 395204; S. F. Tian, F. B. Zhou, S. W. Zhou and T. T. Zhang, Mod. Phys. Lett. B 30 (2016) 1650100. Google Scholar
38. S. F. Tian, J. Differ. Equations 262 (2017) 506; S. F. Tian, Proc. R. Soc. Lond. A 472 (2016) 20160588; P. L. Ma, S. F. Tian and T. T. Zhang, Appl. Math. Lett. 50 (2015) 146. Google Scholar
39. S. F. Tian, Y. F. Zhang, B. L. Feng and H. Q. Zhang, Chinese Ann. Math. B 36 (2015) 543; J. M. Tu, S. F. Tian, M. J. Xu and T. T. Zhang, Appl. Math. Comput. 275 (2016) 345. Google Scholar
40. W. X. Ma and Y. Zhou, J. Differ. Equations 264(4) (2018) 2633; S. T. Chen and W. X. Ma, Front. Math. China 13(3) (2018) 525. Google Scholar
41. J. B. Zhang and W. X. Ma, Comput. Math. Appl. 74 (2017) 591; H. Q. Zhao and W. X. Ma, ibid. 74 (2017) 1399. Google Scholar
42. W. X. Ma, X. Yong and H. Q. Zhang, Comput. Math. Appl. 75(1) (2018) 289; J. Y. Yang, W. X. Ma and Z. Y. Qin, Anal. Math. Phys. 8 (2018) 427. Google Scholar
43. W. X. Ma, Commun. Nonlinear Sci. 16 (2011) 2663. Crossref, Web of Science, ADS, Google Scholar
44. E. T. Be, Ann. Math. 35 (1834) 258. Google Scholar
45. C. Gilson, F. Lambert, J. Nimmo and R. Willox, Proc. R. Soc. Lond. A 452 (1996) 223. Crossref, Web of Science, ADS, Google Scholar
46. F. Lambert, L. Loris and J. Springael, Inverse Probl. 17 (2001) 1067. Crossref, Web of Science, ADS, Google Scholar
47. E. G. Fan and Y. C. Hon, Rep. Math. Phys. 66 (2010) 355. Crossref, Web of Science, ADS, Google Scholar
48. E. G. Fan and Y. C. Hon, Mod. Phys. Lett. B 22 (2008) 547. Link, Web of Science, ADS, Google Scholar
49. E. G. Fan and Y. C. Hon, J. Math. Phys. 53 (2012) 013503. Crossref, Web of Science, ADS, Google Scholar
50. A. R. Seadawy, J. Electromagnet. Waves 31(14) (2017) 1353. Crossref, Web of Science, Google Scholar
51. W. X. Ma, R. G. Zhou and L. Gao, Mod. Phys. Lett. A 24 (2009) 1677. Link, Web of Science, ADS, Google Scholar
52. W. X. Ma, Rep. Math. Phys. 72 (2013) 41. Crossref, Web of Science, ADS, Google Scholar
53. W. X. Ma, Front. Math. China 8(5) (2013) 1139. Crossref, Web of Science, Google Scholar
54. S. F. Tian and H. Q. Zhang, J. Math. Anal. Appl. 371 (2010) 585; J. M. Tu, S. F. Tian, M. J. Xu, P. L. Ma and T. T. Zhang, Comput. Math. Appl. 72 (2016) 2486. Google Scholar
55. S. F. Tian and H. Q. Zhang, Commun. Nonlinear Sci. 16 (2011) 173; X. B. Wang, S. F. Tian, L. L. Feng, H. Yan and T. T. Zhang, Nonlinear Dynam. 88 (2017) 2265. Google Scholar
56. S. F. Tian and H. Q. Zhang, J. Phys. A: Math. Theor. 45 (2012) 055203; M. J. Xu, S. F. Tian, J. M. Tu and T. T. Zhang, Nonlinear Anal.-Real 31 (2016) 388. Google Scholar
57. S. F. Tian and H. Q. Zhang, Stud. Appl. Math. 132 (2014) 212; X. B. Wang, S. F. Tian, M. J. Xua and T. T. Zhang, Appl. Math. Comput. 283 (2016) 216. Google Scholar
58. S. F. Tian and H. Q. Zhang, Chaos Soliton. Fract. 47 (2013) 27; J. M. Tu, S. F. Tian, M. J. Xu and T. T. Zhang, Taiwan. J. Math. 20 (2016) 823; S. F. Tian and H. Q. Zhang, Theor. Math. Phys. 170 (2012) 287. Google Scholar