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Higher-order breathers and breather interactions for the AB system in fluids

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Abstract

In this paper, we perform a systematical study on the breathers of the AB system in fluids. The expressions of the first- to third-order breather solutions are explicitly computed through applying the Darboux transformation method to the system. Some new and interesting features for the breathers are found out as follows. Firstly, in addition to the general breathers, Ak-breathers and Ma-breathers are observed. Further, the corresponding spectrum parameter constrains to form the three breathers are given. Secondly, the dependent variables A and B in the system possess a few of novel spatio-temporal patterns for the periodical wave packets which form the breathers: two-petals, three-petals and four-petals. Thirdly, the interactions between two- or multiple-breathers are elastic. Lastly, there are rich interaction patterns, such as, Ak-breathers and Ma-breather cross vertically, breathers parallel with the same speed.

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References

  1. D. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Nemitz, C. Milford, M.A. Sutton, A two-layer canopy compensation point model for describing bi-directional biosphere-atmosphere exchange of ammonia. Q. J. R. Meteorol. Soc. 127, 815–833 (2001)

    Article  ADS  Google Scholar 

  3. B.F. Farrell, P.J. Ioannou, A theory of Baroclinic turbulence. J. Atmos. Sci. 66, 2444–2454 (2009)

    Article  ADS  Google Scholar 

  4. C.Y. Chen, J.R.C. Hsu, M.H. Cheng, H.H. Chen, C.F. Kuo, An investigation on internal solitary waves in a two-layer fluid: propagation and reflection from steep slopes. Ocean Eng. 34, 171–184 (2007)

    Article  Google Scholar 

  5. M. Chen, K. Chen, Y.X. You, Experimental investigation of internal solitary wave forces on a semi-submersible. Ocean Eng. 141, 205–214 (2017)

    Article  Google Scholar 

  6. D.A. Horn, J. Imberger, G.N. Ivey, The degeneration of large-scale interfacial gravity waves in lakes. J. Fluid Mech. 434, 181–207 (2001)

    Article  ADS  MATH  Google Scholar 

  7. L. Boegman, G.N. Ivey, J. Imberger, The degeneration of internal waves in lakes with slo** topography. Limnol. Oceanogr. 50, 1620–1637 (2005)

    Article  ADS  MATH  Google Scholar 

  8. W. **n, Z.B. Liu, Q.W. Sheng, M. Feng, L.G. Huang, P. Wang, W.S. Jiang, F. **ng, Y.G. Liu, J.G. Tian, Flexible graphene saturable absorber on two-layer structure for tunable mode-locked soliton fiber laser. Opt. Express 22, 10239–10247 (2014)

    Article  ADS  Google Scholar 

  9. P.L.F. Liu, X.M. Wang, A multi-layer model for nonlinear internal wave propagation in shallow water. J. Fluid Mech. 695, 341–365 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. L. Ottolenghi, C. Adduce, F. Roman, G. la Forgia, Large eddy simulations of solitons colliding with intrusions. Phys. Fluids 32, 096606 (2020)

    Article  ADS  Google Scholar 

  11. A. Chesnokov, V. Liapidevskii, Hyperbolic model of internal solitary waves in a three-layer stratified fluid. Eur. Phys. J. Plus 135, 590 (2020)

    Article  Google Scholar 

  12. J.D. Gibbon, M.J. McGuinness, Amplitude equations at the critical points of unstable dispersive physical systems. Proc. R. Soc. Lond. A 377, 185–219 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. A.M. Kamchatnovt, M.V. Pavlovf, Periodic solutions and Whitham equations for the AB system. J. Phys. A Math. Gen. 28, 3279–3288 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  14. R. Guo, H.Q. Hao, L.L. Zhang, Dynamic behaviors of the breather solutions for the AB system in fluid mechanics. Nonlinear Dyn. 74, 701–709 (2013)

    Article  MathSciNet  Google Scholar 

  15. X. Wang, Y.Q. Li, F. Huang, Y. Chen, Rogue wave solutions of AB system. Commun. Nonlinear Sci. Numer. Simul. 20, 434–442 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. X.Y. Wen, Z.Y. Yan, Modulational instability and higher-order rogue waves with parameters modulation in a coupled integrable AB system via the generalized Darboux transformation. Chaos 25, 123115 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. J.J. Su, S. Zhang, Nth-order rogue waves for the AB system via the determinants. Appl. Math. Lett. 112, 106714 (2021)

    Article  MATH  Google Scholar 

  18. L. Wang, Z.Q. Wang, J.H. Zhang, F.H. Qi, M, Li, Stationary nonlinear waves, superposition modes and modulational instability characteristics in the AB system. Nonlinear Dyn. 86, 185–196 (2016)

    Article  Google Scholar 

  19. G.Q. Zhang, Z.Y. Yan, X.Y. Wen, Multi-dark-dark solitons of the integrable repulsive AB system via the determinants. Chaos 27, 083110 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. L. Wang, Z.Z. Wang, D.Y. Jiang, F.H. Qi, R. Guo, Semirational solutions and baseband modulational instability of the AB system in fluid mechanics. Eur. Phys. J. Plus 130, 199 (2015)

    Article  Google Scholar 

  21. G.Q. Meng, H.C. Guo, Mixed solutions for an AB system in geophysical fluids or nonlinear optics. Appl. Math. Lett. 124, 107632 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. J.Y. Zhu, Y. Chen, Long-time asymptotic behavior of the coupled dispersive AB system in low regularity spaces. J. Math. Phys. 63, 113504 (2022)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. R.Z. Gong, D.S. Wang, Whitham modulation theory of the defocusing AB system and its application. Appl. Math. Lett. 126, 107795 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  24. B.L. Guo, L.M. Ling, Q.P. Liu, Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)

    Article  ADS  Google Scholar 

  25. B.Q. Li, Y.L. Ma, Lax pair, Darboux transformation and Nth-order rogue wave solutions for a (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Comput. Math. Appl. 77, 514–524 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  26. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. B.Q. Li, Y.L. Ma, N-order rogue waves and their novel colliding dynamics for a transient stimulated Raman scattering system arising from nonlinear optics. Nonlinear Dyn. 101, 2449–2461 (2020)

    Article  Google Scholar 

  28. B.Q. Li, Y.L. Ma, Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation. Appl. Math. Comput. 386, 125469 (2020)

    MathSciNet  MATH  Google Scholar 

  29. B.Q. Li, Y.L. Ma, The complex short pulse equation: multi-folded rogue waves and phase transition. Appl. Math. Lett. 135, 108399 (2023)

    Article  MATH  Google Scholar 

  30. Y.L. Ma, B.Q. Li, Kraenkel-Manna-Merle saturated ferromagnetic system: Darboux transformation and loop-like soliton excitations. Chaos Soliton Fract. 159, 112179 (2022)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Computer Science and Engineering, Bei**g Technology and Business University, Bei**g, 100048, People’s Republic of China

    Bang-Qing Li

  2. Academy of Systems Science, Bei**g Technology and Business University, Bei**g, 100048, People’s Republic of China

    Bang-Qing Li

  3. School of Mathematics and Statistics, Bei**g Technology and Business University, Bei**g, 100048, People’s Republic of China

    Yu-Lan Ma

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  1. Bang-Qing Li
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Contributions

Bang-Qing Li: Methodology, Formal analysis, Data curation, Writing - original draft. Yu-Lan Ma: Conceptualization, Software, Validation, Writing - review and editing, Methodology.

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Correspondence to Yu-Lan Ma.

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Li, BQ., Ma, YL. Higher-order breathers and breather interactions for the AB system in fluids. Eur. Phys. J. Plus 138, 475 (2023). https://doi.org/10.1140/epjp/s13360-023-04116-9

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  • DOI: https://doi.org/10.1140/epjp/s13360-023-04116-9

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