Skip to main content
SpringerLink
Log in

On the Oceanic/Laky Shallow-Water Dynamics through a Boussinesq-Burgers System

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

Motivation/Development: In order to investigate the shallow-water waves, researchers have introduced many nice models, e.g., a Boussinesq-Burgers system for cetain shallow-water waves near an ocean beach/inside a lake, which we study here via computerized symbolic computation. Originality/Novelty with Potential Application: Concerning the height deviating from the equilibrium position of water as well as the field of horizontal velocity, we now construct a hetero-Bäcklund transformation coupling that system to a known partial differential system, as well as two sets of the similarity reductions, starting at that system towards a known ordinary differential equation. Both our hetero-Bäcklund transformation and similarity reductions lean upon the dispersive power in the shallow water. Results could help the further study on the oceanic/laky shallow-water dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. More fluid-mechanics papers have been published, e.g., in Refs. [18,19,20,21,22,23,24,25].

  2. also called a non-auto-Bäcklund transformation

  3. More computerized symbolic-computation investigations have been reported, e.g., in Refs. [47,48,49,50,51,52,53,54,55,56].

  4. similar to those in Refs. [41,42,43, 58]

  5. Remark 3 of Ref. [58] can help us transform, e.g., a couple of the partial differential equations into a couple of the ODEs.

  6. Similar to those in Refs. [7, 43], for simplicity, we hereby choose a linear function.

References

  1. Dawod, L.A., Lakestani, M., Manafian, J.: Breather wave solutions for the (3+1)-D generalized shallow water wave equation with variable coefficients. Qual. Theory Dyn. Syst. 22, 127 (2023)

    MathSciNet  Google Scholar 

  2. Feng, C.H., Tian, B., Yang, D.Y., Gao, X.T.: Bilinear form, bilinear Bäcklund transformations, breather and periodic-wave solutions for a (2+1)-dimensional shallow water equation with the time-dependent coefficients. Qual. Theory Dyn. Syst. 22, 147 (2023)

    Google Scholar 

  3. Singh, S., Ray, S.S.: New analytic solutions for fluid flow equations in higher dimensions around an offshore structure describing bidirectional wave surfaces. Qual. Theory Dyn. Syst. 22, 123 (2023)

    MathSciNet  Google Scholar 

  4. Gao, X.T., Tian, B., Shen, Y., Feng, C.H.: Considering the shallow water of a wide channel or an open sea through a generalized (2+1)-dimensional dispersive long-wave system. Qual. Theory Dyn. Syst. 21, 104 (2022)

    MathSciNet  Google Scholar 

  5. Gao, X.Y., Guo, Y.J., Shan, W.R.: Symbolically computing the shallow water via a (2+1)-dimensional generalized modified dispersive water-wave system: similarity reductions, scaling and hetero-Bäcklund transformations. Qual. Theory Dyn. Syst. 22, 17 (2023)

    Google Scholar 

  6. Gao, X.Y., Guo, Y.J., Shan, W.R.: Oceanic long-gravity-water-wave investigations on a variable-coefficient nonlinear dispersive-wave system, Wave. Random Complex (2023) in press, https://doi.org/10.1080/17455030.2022.2039419

  7. Gao, X.T., Tian, B., Shen, Y., Feng, C.H.: Comment on “Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system’’. Chaos Solitons Fract. 151, 111222 (2021)

    Google Scholar 

  8. Li, L.Q., Gao, Y.T., Yu, X., Deng, G.F., Ding, C.C.: Gramian solutions and solitonic interactions of a (2+1)-dimensional Broer-Kaup-Kupershmidt system for the shallow water. Int. J. Numer. Method. Heat Fluid Flow 32, 2282 (2022)

  9. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: Shallow-water-wave studies on a (2+1)-dimensional Hirota-Satsuma-Ito system: X-type soliton, resonant Y-type soliton and hybrid solutions. Chaos Solitons Fract. 157, 111861 (2022)

    MathSciNet  Google Scholar 

  10. Zayed, E.M.: Exact traveling wave solutions for a variable-coefficient generalized dispersive water-wave system using the generalized expansion method. Math. Sci. Lett. 3, 9 (2014)

    Google Scholar 

  11. Gao, X.Y., Guo, Y.J., Shan, W.R.: On a Whitham-Broer-Kaup-like system arising in the oceanic shallow water. Chin. J. Phys. 82, 194 (2023)

    MathSciNet  Google Scholar 

  12. Shen, Y., Tian, B., Liu, S.H., Zhou, T.Y.: Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Nonlinear Dyn. 108, 2447 (2022)

  13. Liu, F.Y., Gao, Y.T., Yu, X., Ding, C.C.: Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves. Nonlinear Dyn. 108, 1599 (2022)

    Google Scholar 

  14. Shen, Y., Tian, B.: Bilinear auto-Bäcklund transformations and soliton solutions of a (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves. Appl. Math. Lett. 122, 107301 (2021)

    Google Scholar 

  15. Deng, G.F., Gao, Y.T., Yu, X., Ding, C.C., Jia, T.T., Li, L.Q.: Hybrid waves for a (2+1)-dimensional extended shallow water wave equation. Phys. Fluids 33, 117120 (2021)

    ADS  CAS  Google Scholar 

  16. Gao, X.Y.: Oceanic shallow-water investigations on a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system. Phys. Fluids 35, 127106 (2023)

    ADS  CAS  Google Scholar 

  17. Gao, X.Y.: Thinking of the oceanic shallow water in the light of a (2+1)-dimensional generalized dispersive long-wave system related to HFF 33, 3272; 33, 965 and 32, 2282. Int. J. Numer. Method. Heat Fluid Flow 33, 3801 (2023)

  18. Cheng, C.D., Tian, B., Ma, Y.X., Zhou, T.Y., Shen, Y.: Pfaffian, breather and hybrid solutions for a (2+1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics. Phys. Fluids 34, 115132 (2022)

    ADS  CAS  Google Scholar 

  19. Zhou, T.Y., Tian, B., Shen, Y., Gao, X.T.: Auto-Bäcklund transformations and soliton solutions on the nonzero background for a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in a fluid. Nonlinear Dyn. 111, 8647 (2023)

    Google Scholar 

  20. Feng, C.H., Tian, B., Yang, D.Y., Gao, X.T.: Lump and hybrid solutions for a (3+1)-dimensional Boussinesq-type equation for the gravity waves over a water surface. Chin. J. Phys. 83, 515 (2023)

    MathSciNet  Google Scholar 

  21. Wu, X.H., Gao, Y.T., Yu, X., Ding, C.C.: \(N\)-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics. Nonlinear Dyn. 111, 16339 (2023)

    Google Scholar 

  22. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: Extended (2+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics: solitons, breathers, lumps and interactions. Eur. Phys. J. Plus 138, 305 (2023)

    Google Scholar 

  23. Cheng, C.D., Tian, B., Zhou, T.Y., Shen, Y.: Wronskian solutions and Pfaffianization for a (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation in a fluid or plasma. Phys. Fluids 35, 037101 (2023)

    ADS  CAS  Google Scholar 

  24. Zhou, T.Y., Tian, B., Shen, Y., Cheng, C.D.: Lie symmetry analysis, optimal system, symmetry reductions and analytic solutions for a (2+1)-dimensional generalized nonlinear evolution system in a fluid or a plasma. Chin. J. Phys. 84, 343 (2023)

    MathSciNet  CAS  Google Scholar 

  25. Shen, Y., Tian, B., Cheng, C.D., Zhou, T.Y.: Pfaffian solutions and nonlinear waves of a (3+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt system in fluid mechanics. Phys. Fluids 35, 025103 (2023)

    ADS  CAS  Google Scholar 

  26. Gao, X.Y., Guo, Y.J., Shan, W.R.: Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq-Burgers system for the shallow water waves in a lake or near an ocean beach. Commun. Theor. Phys. 72, 095002 (2020)

  27. Jiang, Y.L., Chen, C.: Lie group analysis and dynamical behavior for classical Boussinesq-Burgers system. Nonlinear Anal. Real 47, 385 (2019)

    MathSciNet  Google Scholar 

  28. Geng, X.G., Wu, Y.T.: Finite-band solutions of the classical Boussinesq-Burgers equations. J. Math. Phys. 40, 2971 (1999)

    ADS  MathSciNet  Google Scholar 

  29. Li, M., Hu, W.K., Wu, C.F.: Rational solutions of the classical Boussinesq-Burgers system. Nonlinear Dyn. 94, 1291 (2018)

    Google Scholar 

  30. Dong, M.J., Tian, S.F., Yan, X.W., Zhang, T.T.: Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq-Burgers equation. Nonlinear Dyn. 95, 273 (2019)

    Google Scholar 

  31. Xu, R.: Darboux transformations and soliton solutions for classical Boussinesq-Burgers equation. Commun. Theor. Phys. 50, 579 (2008)

    ADS  MathSciNet  Google Scholar 

  32. Mei, J., Ma, Z.: N-fold Darboux transformation and multi-soliton solutions for the classical Boussinesq-Burgers system. Appl. Math. Comput. 219, 6163 (2013)

    MathSciNet  Google Scholar 

  33. Zhang, C.C., Chen, A.H.: Bilinear form and new multi-soliton solutions of the classical Boussinesq-Burgers system. Appl. Math. Lett. 58, 133 (2016)

    MathSciNet  Google Scholar 

  34. Liu, W.H., Zhang, Y.F.: Optimal systems, similarity reductions and new conservation laws for the classical Boussinesq-Burgers system. Eur. Phys. J. Plus 135, 116 (2020)

    Google Scholar 

  35. Gao, X.Y., Guo, Y.J., Shan, W.R.: Beholding the shallow water waves near an ocean beach or in a lake via a Boussinesq-Burgers system. Chaos Solitons Fract. 147, 110875 (2021)

    MathSciNet  Google Scholar 

  36. Kumar, S., Rani, S.: Symmetries of optimal system, various closed-form solutions, and propagation of different wave profiles for the Boussinesq-Burgers system in ocean waves Phys. Fluids 34, 037109 (2022)

    ADS  CAS  Google Scholar 

  37. Kumar, R., Pandey, K.S., Kumar, A.: Dynamical behavior of the solutions of coupled boussinesq-burgers equations occurring at the seaside beaches. Braz. J. Phys. 52, 201 (2022)

    ADS  Google Scholar 

  38. Tsiganov, A.V.: Simultaneous separation for the Neumann and Chaplygin systems. Regul. Chaotic Dyn. 20, 74 (2015)

    ADS  MathSciNet  Google Scholar 

  39. Gao, X.Y., Guo, Y.J., Shan, W.R.: Regarding the shallow water in an ocean via a Whitham-Broer-Kaup-like system: hetero-Bäcklund transformations, bilinear forms and \(M\) solitons. Chaos Solitons Fract. 162, 112486 (2022)

    Google Scholar 

  40. Gao, X.Y.: Considering the wave processes in oceanography, acoustics and hydrodynamics by means of an extended coupled (2+1)-dimensional Burgers system. Chin. J. Phys. 86, 572 (2023)

    MathSciNet  Google Scholar 

  41. Gao, X.T., Tian, B.: Water-wave studies on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system. Appl. Math. Lett. 128, 107858 (2022)

    MathSciNet  Google Scholar 

  42. Gao, X.Y., Guo, Y.J., Shan, W.R.: On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water. Nonlinear Dyn. 111, 9431 (2023)

    Google Scholar 

  43. Gao, X.T., Tian, B., Feng, C.H.: Comment on “In oceanography, acoustics and hydrodynamics: An extended coupled (2+1)-dimensional Burgers system” [Chin. J. Phys. 70, 264 (2021)]. Chin. J. Phys. 77, 2818 (2022)

  44. Zhou, T.Y., Tian, B.: Auto-Bäcklund transformations, Lax pair, bilinear forms and bright solitons for an extended (3+1)-dimensional nonlinear Schrödinger equation in an optical fiber. Appl. Math. Lett. 133, 108280 (2022)

    Google Scholar 

  45. Wu, X.H., Gao, Y.T.: Generalized Darboux transformation and solitons for the Ablowitz-Ladik equation in an electrical lattice. Appl. Math. Lett. 137, 108476 (2023)

    MathSciNet  Google Scholar 

  46. Gao, X.Y.: Letter to the Editor on the Korteweg-de Vries-type systems inspired by Results Phys. 51, 106624 (2023) and 50, 106566 (2023). Results Phys. 53, 106932 (2023)

  47. Shen, Y., Tian, B., Zhou, T.Y., Gao, X.T.: N-fold Darboux transformation and solitonic interactions for the Kraenkel-Manna-Merle system in a saturated ferromagnetic material. Nonlinear Dyn. 111, 2641 (2023)

  48. Wu, X.H., Gao, Y.T., Yu, X., Li, L.Q., Ding, C.C.: Vector breathers, rogue and breather-rogue waves for a coupled mixed derivative nonlinear Schrödinger system in an optical fiber. Nonlinear Dyn. 111, 5641 (2023)

    Google Scholar 

  49. Shen, Y., Tian, B., Cheng, C.D., Zhou, T.Y.: N-soliton, Mth-order breather, Hth-order lump, and hybrid solutions of an extended (3+1)-dimensional Kadomtsev-Petviashvili equation. Nonlinear Dyn. 111, 10407 (2023)

  50. Gao, X.Y., Guo, Y.J., Shan, W.R.: Theoretical investigations on a variable-coefficient generalized forced-perturbed Korteweg-de Vries-Burgers model for a dilated artery, blood vessel or circulatory system with experimental support. Commun. Theor. Phys. 75, 115006 (2023)

    ADS  MathSciNet  Google Scholar 

  51. Wu, X.H., Gao, Y.T., Yu, X., Liu, F.Y.: Generalized Darboux transformation and solitons for a Kraenkel-Manna-Merle system in a ferromagnetic saturator. Nonliner Dyn. 111, 14421 (2023)

    Google Scholar 

  52. Shen, Y., Tian, B., Yang, D.Y., Zhou, T.Y.: Hybrid relativistic and modified Toda lattice-type system: equivalent form, N-fold Darboux transformation and analytic solutions. Eur. Phys. J. Plus 138, 744 (2023)

  53. Lü, X., Chen, S.J.: New general interaction solutions to the KPI equation via an optional decoupling condition approach. Commun. Nonlinear Sci. Numer. Simul. 103, 105939 (2021)

    MathSciNet  Google Scholar 

  54. Gao, X.Y., Guo, Y.J., Shan, W.R., Zhou, T.Y.: Report on an extended three-coupled Korteweg-de Vries system. Ricerche Mat. (2023). https://doi.org/10.1007/s11587-023-00769-x

  55. Zhou, T.Y., Tian, B., Shen, Y., Gao, X.T.: Bilinear form, bilinear auto-Bäcklund transformation, soliton and half-periodic kink solutions on the non-zero background of a (3+1)-dimensional time-dependent-coefficient Boiti-Leon-Manna-Pempinelli equation. Wave Motion 121, 103180 (2023)

    Google Scholar 

  56. Shen, Y., Tian, B., Zhou, T.Y., Cheng, C.D.: Multi-pole solitons in an inhomogeneous multi-component nonlinear optical medium. Chaos Silotons Fract. 171, 113497 (2023)

    MathSciNet  Google Scholar 

  57. Bhrawy, A.H., Tharwat, M.M., Abdelkawy, M.A.: Integrable system modelling shallow water waves: Kaup-Boussinesq shallow water system. Indian J. Phys. 87, 665 (2013)

    ADS  CAS  Google Scholar 

  58. Clarkson, P., Kruskal, M.: New similarity reductions of the Boussinesq equation. J. Math. Phys. 30, 2201 (1989)

    ADS  MathSciNet  Google Scholar 

  59. Aksenov, A.V., Kozyrev, A.A.: New reductions of the unsteady axisymmetric boundary layer equation to ODEs and simpler PDEs. Mathematics 10, 1673 (2022)

    Google Scholar 

  60. Ince, E.: Ordinary Differential Equations. Dover, New York (1956)

    Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11871116 and Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11.

Author information

Authors and Affiliations

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

    Xin-Yi Gao, Yong-Jiang Guo & Wen-Rui Shan

  2. College of Science, North China University of Technology, Beijing, 100144, China

    Xin-Yi Gao

Authors
  1. Xin-Yi Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong-Jiang Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wen-Rui Shan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Xin-Yi Gao, Yong-Jiang Guo or Wen-Rui Shan.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, XY., Guo, YJ. & Shan, WR. On the Oceanic/Laky Shallow-Water Dynamics through a Boussinesq-Burgers System. Qual. Theory Dyn. Syst. 23, 57 (2024). https://doi.org/10.1007/s12346-023-00905-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00905-w

Keywords

Mathematics Subject Classification

Search

Navigation