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Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

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Bo Tian

State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

E-mail Address: tian_bupt@163.com

Corresponding author.

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Su-Su Chen

State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

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Cheng-Cheng Wei

State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

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, and

Yu-Qi Chen

State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

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https://doi.org/10.1142/S0217984921504212Cited by:41 (Source: Crossref)

Abstract

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

Keywords:

References

1. S. A. Shan and N. Imtiaz, Phys. Lett. A 383 (2019) 2176. Crossref, Web of Science, ADS, Google Scholar
2. X. Y. Gao, Y. J. Guo and W. R. Shan, Commun. Theor. Phys. 72 (2020) 095002. Crossref, Web of Science, Google Scholar
3. M. S. Osman, D. Baleanu, A. R. Adem, K. Hosseini, M. Mirzazadeh and M. Eslami, Chin. J. Phys. 63 (2020) 122. Crossref, Web of Science, Google Scholar
4. X. Y. Gao, Y. J. Guo and W. R. Shan, Chaos Solitons Fract. 147 (2021) 110875. Crossref, Web of Science, Google Scholar
5. Z. W. Yan and S. Y. Lou, Appl. Math. Lett. 104 (2020) 106271. Crossref, Web of Science, Google Scholar
6. Z. Miao, X. Hu and Y. Chen, Appl. Math. Lett. 112 (2021) 106722. Crossref, Web of Science, Google Scholar
7. J. M. Burgers, Adv. Appl. Mech. 1 (1948) 171. Crossref, Google Scholar
8. U. Obaidullah and S. Jamal, J. Appl. Math. Comput. 65 (2021) 541. Crossref, Web of Science, Google Scholar
9. A. M. Wazwaz and S. A. El-Tantawy, Nonlinear Dyn. 87 (2017) 2457. Crossref, Web of Science, Google Scholar
10. J. D. Cole, Quart. Appl. Math. 8 (1951) 225. Crossref, Web of Science, Google Scholar
11. J. Bec and K. Khanin, Phys. Rep. 447 (2007) 1. Crossref, Web of Science, ADS, Google Scholar
12. V. N. Zyryanov, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 2131. Crossref, Web of Science, ADS, Google Scholar
13. P. J. Olver, J. Math. Phys. 18 (1977) 1212. Crossref, Web of Science, ADS, Google Scholar
14. R. Hirota, Phys. Rev. Lett. 27 (1971) 1192. Crossref, Web of Science, ADS, Google Scholar
15. A. M. Wazwaz, Phys. Lett. A 384 (2020) 126310. Crossref, Web of Science, Google Scholar
16. Y. Shang, Y. Huang and W. Yuan, Appl. Math. Comput. 217 (2011) 7172. Crossref, Web of Science, Google Scholar
17. R. S. Ibrahim and O. H. El-Kalaawy, Phys. Plasmas 13 (2006) 102305. Crossref, Web of Science, ADS, Google Scholar
18. H. A. Zedan, J. Franklin Inst. 348 (2011) 1751. Crossref, Web of Science, Google Scholar
19. R. Hirota, X. B. Hu and X. Y. Tang, J. Math. Anal. Appl. 288 (2003) 326. Crossref, Web of Science, Google Scholar
20. F. Y. Liu, Y. T. Gao, X. Yu, C. C. Ding, G. F. Deng and T. T. Jia, Chaos Solitons Fract. 144 (2021) 110559. Crossref, Web of Science, Google Scholar
21. F. Y. Liu, Y. T. Gao, X. Yu, L. Q. Li, C. C. Ding and D. Wang, Eur. Phys. J. Plus 136 (2021) 656. Crossref, Web of Science, Google Scholar
22. L. Q. Li, Y. T. Gao, L. Hu, T. T. Jia, C. C. Ding and Y. J. Feng, Nonlinear Dyn. 100 (2020) 2729. Crossref, Web of Science, Google Scholar
23. T. T. Jia, Y. T. Gao, X. Yu and L. Q. Li, Appl. Math. Lett. 114 (2021) 106702. Crossref, Web of Science, Google Scholar
24. T. T. Jia, Y. T. Gao, G. F. Deng and L. Hu, Nonlinear Dyn. 98 (2019) 269. Crossref, Web of Science, Google Scholar
25. Y. Shen, B. Tian, C. R. Zhang, H. Y. Tian and S. H. Liu, Mod. Phys. Lett. B 35 (2021) 2150261. Link, Web of Science, ADS, Google Scholar
26. Y. Shen, B. Tian and T. Y. Zhou, Eur. Phys. J. Plus 136 (2021) 572. Crossref, Web of Science, Google Scholar
27. Y. Shen, B. Tian and S. H. Liu, Phys. Lett. A 405 (2021) 127429. Crossref, Web of Science, Google Scholar
28. F. Y. Liu, Y. T. Gao, X. Yu, L. Hu and X. H. Wu, Chaos Solitons Fract. 152 (2021) 111355. Crossref, Web of Science, Google Scholar
29. D. Wang, Y. T. Gao, C. C. Ding and C. Y. Zhang, Commun. Theor. Phys. 72 (2020) 115004. Crossref, Web of Science, ADS, Google Scholar
30. D. Wang, Y. T. Gao, J. J. Su and C. C. Ding, Mod. Phys. Lett. B 34 (2020) 2050336. Link, Web of Science, ADS, Google Scholar
31. D. Wang, Y. T. Gao, X. Yu, L. Q. Li and T. T. Jia, Nonlinear Dyn. 104 (2021) 1519. Crossref, Web of Science, Google Scholar
32. D. Y. Yang, B. Tian, Q. X. Qu, C. R. Zhang, S. S. Chen and C. C. Wei, Chaos Solitons Fract. 150 (2021) 110487. Crossref, Web of Science, Google Scholar
33. L. Hu, T. Gao, S. L. Jia, J. J. Su and G. F. Deng, Mod. Phys. Lett. B 33 (2019) 1950376. Link, Web of Science, ADS, Google Scholar
34. D. Y. Yang, B. Tian, Q. X. Qu, Y. Q. Yuan, C. R. Zhang and H. Y. Tian, Mod. Phys. Lett. B 34 (2020) 2150013. Link, Web of Science, ADS, Google Scholar
35. D. Y. Yang, B. Tian, H. Y. Tian, Y. Q. Chen, W. R. Shan and Y. Jiang, Optik 247 (2021) 166815. Crossref, Web of Science, ADS, Google Scholar
36. L. Liu, B. Tian, H. L. Zhen, D. Y. Liu and X. Y. Xie, Wave Random Complex 28 (2018) 343. Crossref, Web of Science, ADS, Google Scholar
37. H. Zhao and W. X. Ma, Comput. Math. Appl. 74 (2017) 1399. Crossref, Web of Science, Google Scholar
38. C. C. Ding, Y. T. Gao, G. F. Deng and D. Wang, Chaos Solitons Fract. 133 (2020) 109580. Crossref, Web of Science, Google Scholar
39. C. C. Ding, Y. T. Gao and G. F. Deng, Nonlinear Dyn. 97 (2019) 2023. Crossref, Web of Science, Google Scholar
40. G. F. Deng, Y. T. Gao, C. C. Ding and J. J. Su, Chaos Solitons Fract. 140 (2020) 110085. Crossref, Web of Science, Google Scholar
41. G. F. Deng, Y. T. Gao, J. J. Su, C. C. Ding, T. T. Jia, Nonlinear Dyn. 99 (2020) 1039. Crossref, Web of Science, Google Scholar
42. J. J. Su, Y. T. Gao, G. F. Deng and T. T. Jia, Phys. Rev. E 100 (2019) 042210. Crossref, Web of Science, ADS, Google Scholar
43. J. J. Su, Y. T. Gao and C. C. Ding, Appl. Math. Lett. 88 (2019) 201. Crossref, Web of Science, Google Scholar
44. Y. J. Feng, Y. T. Gao, L. Q. Li and T. T. Jia, Eur. Phys. J. Plus 135 (2020) 272. Crossref, Web of Science, Google Scholar
45. Y. J. Feng, Y. T. Gao, T. T. Jia and L. Q. Li, Mod. Phys. Lett. B 33(29) (2019) 1950354. Link, Web of Science, ADS, Google Scholar
46. M. Wang, B. Tian, Y. Sun and Z. Zhang, Comput. Math. Appl. 79 (2020) 576. Crossref, Web of Science, Google Scholar
47. M. Wang, B. Tian, C. C. Hu and H. Liu, Appl. Math. Lett. 119 (2021) 106936. Crossref, Web of Science, Google Scholar
48. M. Wang, B. Tian, Q. X. Qu, X. H. Zhao and C. R. Zhang, Mod. Phys. Lett. B 35 (2021) 2150020. Link, Web of Science, ADS, Google Scholar
49. M. Wang, B. Tian, Q. X. Qu, X. H. Zhao, Z. Zhang and H. Y. Tian, Int. J. Comput. Math. 97 (2020) 2474. Crossref, Web of Science, Google Scholar
50. M. Wang, B. Tian, S. H. Liu, W. R. Shan and Y. Jiang, Eur. Phys. J. Plus 136 (2021) 635. Crossref, Web of Science, Google Scholar
51. M. Wang and B. Tian, Eur. Phys. J. Plus 136 (2021) 1002. Crossref, Web of Science, Google Scholar
52. A. M. Wazwaz, J. Franklin Inst. 347 (2010) 618. Crossref, Web of Science, Google Scholar
53. A. V. Tsiganov, Phys. Lett. A 379 (2015) 2903. Crossref, Web of Science, ADS, Google Scholar
54. X. Y. Gao, Y. J. Guo and W. R. Shan, Phys. Lett. A 384 (2020) 126788. Crossref, Web of Science, Google Scholar
55. X. Y. Gao, Y. J. Guo and W. R. Shan, Qual. Theory Dyn. Syst. 20 (2021) 87. Crossref, Web of Science, Google Scholar
56. X. Y. Gao, Y. J. Guo and W. R. Shan, Appl. Math. Lett. 120 (2021) 107161. Crossref, Web of Science, Google Scholar
57. X. Y. Gao, Y. J. Guo and W. R. Shan, Wave. Random Complex (2021) in press. https://doi.org/10.1080/17455030.2021.1942308 Google Scholar
58. X. Y. Gao, Y. J. Guo and W. R. Shan, Rom. Rep. Phys. 73 (2021) 111. Google Scholar
59. A. V. Taiganov, Regul. Chaotic Dyn. 20 (2015) 74. Crossref, Web of Science, ADS, Google Scholar
60. X. T. Gao, B. Tian, Y. Shen and C. H. Feng, Chaos Solitons Fract. 151 (2021) 111222. Crossref, Web of Science, Google Scholar
61. Y. Shen and B. Tian, Appl. Math. Lett. 122 (2021) 107301. Crossref, Web of Science, Google Scholar
62. L. Hu, Y. T. Gao, T. T. Jia, G. F. Deng and L. Q. Li, Z. Angew. Math. Phys. 72 (2021) 75. Crossref, Web of Science, Google Scholar
63. Y. Shen, B. Tian, X. Zhao, W. R. Shan and Y. Jiang, Pramana-J. Phys. 95 (2021) 137. Crossref, Web of Science, ADS, Google Scholar
64. Y. X. Ma, B. Tian, Q. X. Qu, C. C. Wei and X. Zhao, Chin. J. Phys. 73 (2021) 600. Crossref, Web of Science, Google Scholar
65. Y. X. Ma, B. Tian, Q. X. Qu, D. Y. Yang and Y. Q. Chen, Int. J. Mod. Phys. B 35 (2021) 2150108. Link, Web of Science, ADS, Google Scholar
66. Y. X. Ma, B. Tian, Q. X. Qu, H. Y. Tian and S. H. Liu, Mod. Phys. Lett. B 35 (2021) 2150315. Link, Web of Science, ADS, Google Scholar
67. H. Tasso, J. Phys. A 29 (1996) 7779. Crossref, ADS, Google Scholar
68. Z. Q. Gu, J. Math. Phys. 31 (1990) 1374. Crossref, Web of Science, ADS, Google Scholar
69. A. H. Chen, Phys. Lett. A 374 (2010) 2340. Crossref, Web of Science, ADS, Google Scholar
70. B. Xue and C. M. Wu, Commun. Theor. Phys. 58 (2012) 317. Crossref, Web of Science, ADS, Google Scholar
71. M. Hamanaka and K. Toda, J. Phys. A: Math. Gen. 36 (2003) 11981. Crossref, ADS, Google Scholar
72. H. Y. Tian, B. Tian, C. R. Zhang and S. S. Chen, Int. J. Comput. Math. (2021) in press. https://doi.org/10.1080/00207160.2021.1900568 Google Scholar
73. H. Y. Tian, B. Tian, Y. Sun and C. R. Zhang, Commun. Nonlinear Sci. Numer. Simul. (2021) in press. https://doi.org/10.1016/j.cnsns.2021.106097 Google Scholar
74. L. Q. Li, Y. T. Gao, X. Yu, T. T. Jia, L. Hu and C. Y. Zhang, Chin. J. Phys. (2021) in press. https://doi.org/10.1016/j.cjph.2021.09.004 Google Scholar
75. M. Wang, B. Tian and T. Y. Zhou, Chaos Solitons Fract. 152 (2021) 111411. Crossref, Web of Science, Google Scholar
76. M. Wang and B. Tian, Wave. Random Complex (2021) in press. https://doi.org/10.1080/17455030.2021.1986649 Google Scholar
77. C. C. Ding, Y. T. Gao, L. Hu, G. F. Deng and C. Y. Zhang, Chaos Solitons Fract. 142 (2021) 110363. Crossref, Web of Science, Google Scholar